So, how do you know that a man really loves you? Is it by the grand gestures that he makes when he proposes, or is it by the strong eye contact that he makes in romantic situations? Or is it the way that he treats you like a queen around his friends and his family? And how do you know that he's not faking it? Today, we're going to be talking about eight things the man will only do if he really loves you. My name is Matthew Coast, and welcome to Commitment Connection. If you're, before we get started, if you're really serious about getting into a relationship where you feel loved and cherished and treated like a queen, make sure that you go to thegoddaskommunity.com and consider joining our community there, and it's really awesome. So check it out. So how do you know that a man really loves you? What are some of the things that he will do? So number one is he will love to make you laugh. And so the reason that this one is important is because he's coming from a space where he wants to please you. If you're watching this right now and you're in the chat, make sure that you say hello and tell us where it is that you're watching this from because it's really awesome to see how many people from all over the world are watching this. So a man, if a man is in a space where he really truly loves you, he's going to want to please you. He's going to want to make you happy. He's going to want to make you feel good. He's going to want to do things that makes you feel like you enjoy being around him. And so one of the big ways that you can tell that he's doing this is he's going to be telling jokes. He's going to make little sarcastic comments. He might even do like little funny faces or something weird, right? And what he's trying to do is trying to get you to laugh because that's like music to his ears.  He's trying to impress you by performing. And one of the ways that he knows that it's working is that you're laughing. And so that's number one. So number two is that he's going to make time for you. And what is this?  And so if a man is really, really in love with you, he's going to want to make you a priority.  He's going to go out of his way to come and spend time with you. He's going to drive across the city. He's going to do whatever he has to do. He's going to move mountains. He's going to do whatever he has to do. He's going to fly to the other end of the earth, right? Those are the cliches out there about doing this. So he's going to make time for you.  And so all of these things, what you want to notice is how it happens over time. And one of the other things that you want to notice too is that being in love, which is what we're talking about here, we're not talking about the act of loving somebody, which is something a little bit different, right? It's a choice. It's a thing that you do for somebody. We're talking about this emotional feeling of being in love, right? Falling in love. And being in love is something that can grow. It can happen more. It's something that can be smothered. It's something that can happen less. It can be pulled away. And so what you want to kind of pay attention to here is kind of building this and where is he? Like where is he on this scale of like he's really into you? Or he's not really that into you. And so if his interest and desire, if his feelings of being in love kind of drop below a certain point, he's going to stop doing all these things and eventually he's going to start pulling away and he's going to lose interest and then he's going to disappear on you. And that's how you know. And if he goes the other direction, right, he's going to become more interested in you. He's going to start doing these things more and more and more. You're going to become more of a priority. You're going to become more of all the things that we're going to be talking about here. And so that's what you want. That's what you want to pay attention to. And at the end, we'll talk a little bit more about how that feeling of in love actually happens and how to build that and how to create that in your man. And so number three, number one was he will love to make you laugh. Number two is he makes time for you. Number three is that he introduces you to his friends or his family or both, depending on how he feels about his friends and his family. Even if it's not like a big deal, right? And so what do we mean by that, right? Passing, there's kind of this test where guys will take you around his family and he'll take you around his friends and he's kind of like showing you off and he's letting everybody know that he has this amazing person that he really cares about in his life and that he's proud to have. And that's why he shows you to his family and his friends is because he's proud. He feels like he's done a great job.  And I'm showing her off to my friends and family. And I want them to see how great of a job that I've done here so far. And so one of the aspects of that is that he's going to show you to his friends and his family. And so it's, you know, is he doing that, right?  Like if you're like, oh, you want me to meet your friends and your family, right? And for a lot of women, it's like, whoa, like what's going on here, right?  Like it's not a big deal, even though it really is a big deal for him. He might try to say like it's not a not a big deal or he might try to quell him that because he wants to make it, wants you to feel comfortable. He wants to make it a good experience and he wants to, he wants to do it. And so he'll, he'll do that and he'll bring you around his friends and his family. And he might try to kind of chill, chill everything out and make it feel kind of more relaxed and more comfortable doing that with you so that you feel comfortable and you make a good impression and everybody likes each other. And sometimes that doesn't happen though, right? Like if men have families where they don't like their family or they don't care about their family or they were really hurt by their family as a child or their, their parents weren't really there for them or something. Sometimes guys will kind of develop other kind of family. Like their friends will become their family or something like that. And so it kind of depends on what's going on with the guy that you're seeing and how he feels in that regard as well. So that's just something to make a note of. So number four is he will make small but significant changes without being asked. And this is the concept that I've talked about before where I talk about the differences between force versus power, right?  Whereas coming from a space of power is when a guy chooses this stuff. When you're coming from a place of power, men will choose to do this. They will, they will decide that they want to step up and start making changes in themselves in order to impress you and to make you feel good and to show you that they really care about you by doing these little things. Because he'll hear things and he'll be like, oh, she said that. And this is something that I need to do. And he'll think about it and he'll be like, yeah, but I really care about her. And I want her to see how awesome I am and feel like I'm an awesome man. And so he'll do a lot of those things. And so like I said, a man will not want to change for a woman who tries to force him or who puts pressure on him to change his behavior. It's really gross. It feels gross for a guy. If a woman is kind of coming from this place of force where she's trying to manipulate him, she's playing games with him. She's doing all these things to try to get him to do these things. She's nagging him or whatever, right? He's going to want to fight back against it. And so, but if he really loves you and you're coming from a space of power, then he'll want to do those things on his own. If you get what I'm saying right now in the chat, say, I get it. Say, I get it. If you get what I'm talking about here, say, I get it.   And that's what a guy who's really in love will do.  Whatever the weird things are that you do, you have a little weird dimple somewhere or you've got a freckle here or there, whatever, right? Those things will become these things that he starts loving about you.  And what you should be doing when he does that is just opening yourself up to receive, just accept it in, just allow it to come in and allow whatever he's saying, whatever he's doing to just come, just allow it to come, right? Just allow yourself to receive it. So number six is he'll become very, very protective over you, right? So one of the things that men do is we're providers and protectors. Maybe you've heard that before and men will want to protect you, right?  He's going to want to continually always protect you and make sure that you're okay. If you have some people that are, you have a complaint about something that's going on and you're telling them about that, he'll want to fight those people, right? Like that's the sign that he's really into you and that he really loves you is that he wants to protect you and you might be like, whoa, whoa, whoa, the way we don't need to move to violence here, right?  And so that's, it's a big sign that he's in love with you. He has those feelings of being in love with you. And so number seven is he'll always make time to listen, right? He'll try to get present, he'll try to listen to what it is that you're doing. And a lot of players you'll notice are, they hate doing this, right? They hate listening because they don't really care that much in reality, is a lot of players, they don't care.  And if he's really in love, he'll want to listen to you. He'll want to care about what it is that you're saying. He'll want to listen to all of the sad things that you have to say, your anger, your funny stories, the mundane things that you have going on in your life. You know, if he's in love with you, he'll want to have conversations with you about this stuff. It'll feel less like a chore to him, the more in love he is with you. And it'll feel more like a privilege to experience your life with you and listen to these things that are going on with you. And a lot of women sometimes, if they're feeling insecure or whatever, they'll be like, oh, he doesn't care or whatever, right? Or I don't want to burden him with these things. And a guy that's really in love, well, it won't be a burden. It'll be something that he really wants to experience and he really wants you to talk about. So number eight is you are in his future. And so if a guy's really in love with you, he's going to think about you in the future. And he's going to be like, oh, I want to have this woman in my life in the future. He'll start planning things. And sometimes, and I'm just going to throw a caveat in here. I'm going to throw in in a little instance where this might kind of not happen. And it kind of depends on if he's heard any men's dating advice or not. Because a lot of times in the men's dating industry, what they'll tell people is not to do a lot of these things, right? Because it shows that you're interested. And if you show a lot of guys out there, just like in the women's dating industry where you'll get some of that advice where people will be like, oh, don't show interest in him or whatever, right? They tell the same thing over to the guy's industry. And so he might try to not get overly excited about it, right? And try to plan things with you because he's like, oh, I don't want to show too much interest and scare this woman away or whatever when he's really kind of falling in love with you and he really wants these things.  He'll try to plan trips with you. He'll try to plan things to do with you in the future. Like go to concerts or do different kinds of things. He'll want to just have you in his life as much as he possibly can in terms of trying not to like be too pushy about it and not freak you out because he's in love with you. And he's not, you know, it's hard when you're in love with somebody because you look at them and sometimes you kind of like project your feelings onto that person, but you may know from your past experiences that sometimes what you feel is not what that person feels.  They're like, oh, you know, and you can end up smothering them and pushing them away if you're kind of showing it too much. And it's really not even showing it too much, but it's smothering people that actually does that. And a lot of times people don't know that. And so sometimes guys will try to pull back on it, but you'll see, you'll end up seeing glimpses of it, which is why we're talking about the things that a man will only do if he really loves you. And so those are the eight things. Number one is he loves to make you laugh. Two is he makes time for you. Three is he will introduce you to his family, even though it's not a big deal. Four is he will make small but significant changes without being asked.  Six is he'll become very protective over you. Seven is he will always make time to listen. And eight is he will want you in his future. So again, if you want to, if you're really serious about having the relationship that you've always wanted to have, make sure that you go over to the goddesscommunity.com and consider joining our community there for free.  And so go and check that out. And okay, so what else? All right, I was gonna talk, I said I'd talk a little bit about what makes a man fall in love. And so there's certain kind of levels of a man's feelings, right? So when you meet a guy, if he's really into you and his interest level is fairly high, he will start to develop feelings of attraction and it'll build and build and build and eventually grow into this thing of feeling in love if you allow it to grow and you give it space to grow but you don't give it so much space that it kind of putters out because it wasn't giving enough attention. So the metaphor is the fire, right?  And eventually what ends up happening is you can throw logs and stuff on it and it won't smother and it'll just get bigger and bigger and bigger and bigger and at the same time, you can end up smothering it if you do too much, right? Anybody that knows that's gone out and actually tried to build a fire before knows that you can end up smothering a fire completely and destroying it and so you don't wanna do that. And what you do wanna do is give it room to breathe, right? Like love isn't something that you can really force. It's something that you can allow to happen, right? And the way that it happens is by connecting with it and giving to it, but also giving it space so that it can build and grow and having that connection there, building that deeper connection from a space for women, typically from the space of being feminine, from connecting to your own emotions and then connecting to him and allowing that kind of interest to grow and build and get deeper and to expand. It's something that I call the emotional range principle that I'm actually right now, I'm recording the Forever Woman program that I'm creating for our community that I'm gonna be giving away as part of our community for free. And so if you wanna check that out again, thegoddiscommunity.com if you've got any questions or anything, I'm gonna go through the chat here real quick and I'm going to see what everybody's talking about. If you have any questions about what we just talked about or if you have any questions about anything that's going on in your own particular situation, throw them into the comments section, throw them into the chat and I will get to them right now. So we've got lots of good mornings. Good morning, I'm not sure. It depends on where you are in the world, I guess whether it's morning or it's afternoon or evening or night. So hello, hello, hello everyone. Sissy, Kelly, New York, Columbus, Ohio, nice. London, Michigan, Canada, Taiwan, Malaysia, awesome Philippines, New Orleans. Upper tar, upper tar says hello. Brie Mae says watching out of depression from being cheated, more like betrayed by my boyfriend. I don't even get that he loved me. It's very complicated. Well, I'm sorry to hear that, Brie. So New Zealand, Australia, all right. Everybody sounds like lots of people are getting it. Lots of people are getting it. Cathy says, thanks Matthew for the awesome advice. You're welcome, welcome Cathy. So Infinite Queen B says yes, he puts his arm over to protect me. Yes, exactly. From you jerking forward when another car made us stop suddenly. Yes, exactly what I'm talking about here. Met Monk, how you doing Met Monk? She said she doesn't get notifications of live streaming anymore. Did you check your, so make sure that under, if you're new to our channel, make sure that you subscribe and there's a bell notification next to the subscribe button. And you have to click on that if you want to get notified. Sometimes for some reason, YouTube doesn't notify everybody.  You won't be confused and frustrated about where you stand. That's right. Germany, Tricia says she's from Germany. Barbara says, we met on a dating site and have been together for two years. He has now started doing all these things and wants to be more with me more and more awesome. Congratulations, Barbara. That is so cool. Yeah. Rachel says, go get Bre May.  It's true. I mean, it's one of those things. And one of the things that I talk about sometimes is this idea of abundance. And so having more options and having more people, like one of the big kind of problems that a lot of times happens with women that are in our community is that they'll get so fixated and focused on one guy and they will give everything to that guy and they will end up, that guy will become their world and their worlds will be kind of mixed together. And what ends up happening is that the woman's identity starts getting mixed together with the guy's identity.  And so there's this big kind of vortex space and she kind of gets sucked into it.  It feels like somebody that's going through like withdrawal symptoms from using drugs.  And if you create abundance, if you create a life that's your own, if you make sure that you have your own identity in addition to having, being connected to a man, then it, one, that's far less likely to happen because you're creating space and you're allowing, like I said, that fire to grow instead of smothering it. And two, if something does end up happening, then it's a lot less painful for you. And it's, and if you have kind of new guys and circulation and stuff and you're contacting them and you're getting in touch with them and you're dating them, it actually can end up feeling really good. And all of a sudden you're like, whoa, you know, like when I was so stuck on what that one person was doing, it was really painful. And now that I've opened myself up and there's all these other people and there's these guys who are trying to court me and they're pursuing me and they're chasing me, all of a sudden I don't feel so bad when this one person is misbehaving because it's not a big deal.  You can take that fish. I don't know if any of you have ever fished before but I've done some fishing before.  You know, if you're ready to behave but if not, you can just hang out in the pond or the ocean or whatever you wanna call them until you get it together. That's what you need to do is get it together. So Mrs. Paranoid, Android says, Matt, I have a question because you once said not to have him move in with you unless he puts a ring on your finger. What did you mean with that and why is marriage so important? Well, what I mean is if you want marriage which you may not want, a lot of people don't want marriage anymore which is fine, there's nothing wrong with not wanting marriage. But if you want marriage, that's very important because what's gonna end up happening is the guy is going to get all of his needs fulfilled. And he's gonna be like, oh, this is, I don't need to do anything more because I've already got this girl and she's with me and all of my needs are being fulfilled so I don't feel like I need to kind of show her this thing of love where I get her a ring and do all this stuff because I've already reached the end goal. A lot of times guys look at these things in terms of goals and they're like, okay, this is what I'm looking for. And once he hits that goal and you'll hear this from a lot of women is when they move in with guys what ends up happening is the guy starts getting lazy. And that's a number of different problems together. One of the problems is that he's already reached his goal. Another one of the problems is that whole thing about the scarcity principle which I talk about which is that you're around him all the time and so he starts taking you for granted because you're always there and you're living with him now instead of being somebody that he appreciates more because of the contrast of you being there versus not being there which is what makes somebody appreciate something that they have is the experience of it not being there. If you have something all the time, so for instance, let's talk about health.   It's so amazing. And you appreciate not being sick, right? Which is one of those things where it sucks, it kinda sucks that it takes removing something in order for somebody to appreciate it but it's just human biology, right? This is something, human psychology. It's something that we appreciate when things, when we experience the contrast of it being there versus it not being there. And it's always that way and there's not a whole lot you can do about it other than make sure that you give space in order for something to appreciate it. So a dear Ramee says thank you for your advice, you're welcome. Yeah, Rachel says, yep, who cares if he cheated with a million girls you can go get a million other men too, boom, you have to know who you are. All right. So Rebecca says, hi, I'm Rebecca from Nottingham. I am in a relationship but me and my fiancee are keeping our relationship a secret because his friends think I am too childish. We haven't been talking much. Yeah, I mean, that's a problem. One of the things that I always say is that rule number one is that you never wanna be a part of a relationship that has to be kept a secret. And just realize that if he feels that, if he actually feels that way about you and he feels that way about you for a long period of time, I don't know how long you've been. You and your fiancee. I mean, it's, so you're getting married to a guy that wants to keep you a secret from his friends. I mean, that's just weird, you know? Like, how is that gonna play out when you're married?   That's just what it is. So Metmonk says, just met a nice man, been on a first date, next date, will have to be in seven weeks. I am going to walk the Camino in Spain. Wonder if it can last on these conditions. Seven weeks, huh? That's, it's a long time. That's a long time. So you went on a first date and then the next date will be in seven weeks. Oh, yeah. I mean, I wouldn't have too high of expectations for that. I mean, so many things can happen in seven weeks when you're gone, because that's what we're talking about in terms of like the fire, right? Like it's, you're completely removing yourself from it and it can completely die or not. I mean, it just really kind of depends. Seven weeks is a long time though between a first and a second date. Elizabeth says, what if he does this and then pulls away off and on? Well, it depends on what you mean by pulling away. Sometimes women will say he pulls away, but what she really means is that we haven't talked in two days. And so it kind of depends on what you mean by pulled away because, you know, having space is a normal thing, right? You want to give him space before he needs it so that you can allow, like what we're talking about, the fire to build. And if, and it depends on what kind of a relationship you have. Are you guys talking all the time? Are you talking every day? You know, what does it mean when he pulls away off and on? So Annalisa says, I love your advice. Thank you, watching from the Philippines. Thank you, I'm glad that you loved my advice. Milanka says, watching you from Western Australia, cool. Annalisa says, how are you single? No, I'm not. Wiggy Girl says, my boyfriend has already moved in. So that's it. Well, I would just remember that a couple of different things, Wiggy Girl, is that first off, what you have to know is that a guy knows what kind of a relationship that he wants to have with you by the six month mark. So I don't know how long it's been since you guys have started dating, but after by the six month mark, he knows whether he wants to get married to you, whether this is as far as it's going or what's going on, right? And so you're in a situation where you moved in and the question is, do you want a rent? Do you want to get married? Do you want that? And so the pressure that happens with a guy is that you end up kind of being this person where you're like, okay, what is it that I really want? And so if one of your needs that you know is that you want to get married and you want to have a marriage, then it's something that, one, you need to talk about with them. You need to have communication with him and you need to have a conversation with him to make sure that it's something that he wants, especially if you're moving in together. This is where you're at the point where you need to be having real serious conversations about what the two of you want and where this is going. And if you're not having those conversations, then you're just leaving it up to chance and those kind of conversations will tell you whether you should be moving in with him and where this is gonna go and whether it's gonna ultimately lead to the type of relationship that you want to have. And like I said, one of the most important things that you can possibly do, I've said this in other live streams. I talk about this in the forever woman program is that your mindset has to shift here from going from this place of being like, this man is who I want to this relationship is what I want. And so having an idea of what kind of relationship that you want is vastly more important than hanging onto a guy because the guy might not want to have that relationship. He might not be willing to give you that relationship. And if you're willing to walk away and you're willing to kind of open things up for yourself and you're focused on the relationship as being something that's more important than the guy that you want to have is more likely to provide that type of relationship because you're coming from a space of power instead of a space of force. Sorry, it just started like raining crazy outside my window here right now. So we get that here sometimes. We get that here sometimes.  Maybe you will, maybe you will. So the hero pilgrim in Spain, I was actually thinking about going to Spain here in what month is it right now? Is it July or in July, August? So I was thinking about going to Spain here in September. July, August, September. So September going out to the, there's some cool beaches out there that I was thinking about checking out. So Melanka says, how old are you? I'm 37. Rachel says, it's like some adults that live with parents, they don't see the need to move out because mom and dad are providing.   Yeah, I mean, it's the whole thing, right? So men, it's not just about sex, right? A lot of times women kind of focus on that piece because they know that it's something that guys really, really like. There's a saying out there that goes that men need sex in order to feel love and women need love in order to want sex. And I don't fully agree with that but it's kind of an interesting thing, right? So guys kind of feel this pressure around the inside themselves biologically in order to have sex. And it's one of those things where, so they a lot of times push for it and also gives a man more power in a relationship once they pass that physical intimacy point. And so men tend to focus a lot on that and think about it a lot. And a lot of times women kind of think about that but there's also emotional needs that men have. There's a lot of emotional needs that men have where they, where those are also something that if you're fulfilling all of those emotional needs that's also something that will make a man lazy and feel like he doesn't need to do these things because he's getting all of those needs fulfilled as well. And so yeah, it's something that you want to think about. So Mayet says he always makes me laugh of his quirky jokes, but is it normal that he shared our funny experiences to his friends and laugh about it? Yeah, I mean, it's one of those things. I was actually just talking to this men's dating advice coach the other day and we were talking about, he was telling me about this thing where he's like, yeah, women don't want you to share information about what's going on between you two with other people because it ends up getting around back to her and it's like this weird thing where she'll go out and talk to all of her friends and they'll gossip about what's going on with their guy but they don't like men doing that with them and is it normal? Yeah, it's normal. I mean, it's a normal thing and I mean, it depends on how you feel about it and if it's something where you don't like it and it feels like it feels really gross to you or it feels kind of weird to you, it's something that maybe you should talk to him about and kind of communicate how you feel about it and just see how he, like what he says and how he responds to that in reference to doing that. So Trisha says, what can I do after he blocked me online? So he blocked you online. When somebody blocks you online, it's a hint about what has been going on, right? Like you have been violating somebody's space to the point where they feel like they need to block you. That should be a signal to you that says that you have been acting in a way that is not appropriate for the situation and you need to back off and chill and let go and lean back and take some time off and get yourself out of the situation, go create abundance and do something else. If a guy's blocking you online, you need to chill.  Part of it, secret I would pick up and deny I even dated him, yeah, yeah, yeah, yeah. We just talked about that actually. So Wiggy Girl says, we've been dating for seven months and he keeps talking about marriage. I get on very well with his children and his eldest daughter even said we should get married. He brings up marriage all the time. Well, that's a good sign, that's a really good sign. Rachel says men put girls in categories, that's how God created it. Adam was told by man to name all of the creatures on earth so men always put women in situations and categories. Okay, Liz says good morning from California.  All right, so Rachel says it's up to woman to decide which category she wants to be placed in. That's right, it does, it is up to you. It is up to you to make sure that you're not settling for being put in a type of relationship that you don't wanna be in and that's what we talk about on this channel and what I'm talking about in my forever woman program and in my community is making sure that you're empowering yourself so that you don't end up settling for something less and you don't end up accidentally falling into something less than what you feel like you deserve and that you want for that matter. Margaret says from Queensland don't know what he wants. Melanka says I was 37, 20 years ago. Wow, well, I am still your older brother. I am here giving you advice as your older brother that I don't know, it's an inside joke I guess that we have here that I'm the older brother of dating advice for all you ladies in the audience. Ferries in Unicorn says hi from the Philippines. Thank you for all you do. Thank you very much for saying that you're welcome. For Florencia says hi, could you explain the idea of feminine and masculine energy in a relationship? Well, that is a massive topic, feminine and masculine energy in a relationship and I'm going to do an entire video on that here pretty soon. Let's see if there's something that I specifically, is there anything specifically that you're looking for or you just feeling like you're being too masculine in your relationship? The whole point is to create polarity, right?  I could talk about that all, literally all day long. I've got sheets, I've created sheets on it. I've got diagrams, which is a very masculine thing to do. I've got diagrams, I've got boxes, I've got football kind of diagrams where you're like, all right, go here and you throw it and then you run that direction, right? And you come back around and I don't know, I'm just making things up at this point. But what do you wanna know about feminine and masculine energy in a relationship? Melanka says, Trish, if he blocked you online just forget about him, walk away, that is best. Shan says, good morning in Canada. What happens if the man is just very guarded? So what happens? What happens if he's just very guarded? I don't know, what are you trying to get at with that, right? He's very guarded and what? You're trying to get him to open up. You want him to communicate with you. You want him to get emotional with you. Like what is your thing that you're trying to do here? Trisha says, thank you, Matt. Yeah, you're right. I think he doesn't agree with my thoughts. Anyway, I'm always willing to walk away. Not sure what thoughts those are that he doesn't agree with, but it looks like we've run at the end of all the questions. So how do you know if he truly cares? That's kind of what I just talked about in this livestream so you wanna go back and watch the video from the beginning. So again, if you're really serious about getting into the relationship that you've always wanted, make sure that you go to thegoddaskommunity.com and consider joining our community there.  You can join our community with thousands of women online and you can get lots of all kinds of stuff that will help you create the relationship that you've always wanted to have. So go check that out. And thank you everybody for being with me here today. It's always a pleasure to interact with you and it's an honor for you, for me to have you allow me to be a part of your journey for creating the life that you want, for creating the relationship that you wanna have. So thank you so much for being here and I will speak with you again soon.


I don't try the exercises or anybody has the collection of exercises because I realized that in the web page is not, well, I guess everybody has seen this, no, but so in the web page here, you have when here when it says the slides, you have the link to the lesson one slides are the slides, but then there is also the outline actually in the outline you can also find the some references. The course is based on these three reviews. This is a review that we wrote in Nature Physics and it's more a kind of overview of the whole topic. This one that you can download from archive is more or less the course except for the information flows which is very briefly discussed in this paper. And this one, information flows in nanomachines is a paper, it's a review but pedagogical. These three papers, they try to be pedagogical, but they try to be pedagogical, I mean easy to read or at least self-contained. Well, this is not easy to read the whole thing. And the part of information flows which is something that I think is interesting here is more advanced but it is something that it is now people is using a lot to understand molecular motors in biology and nanomachines to try to interpret those machines as Maxwell Demons. So this is what we will study next week and here you have its base on this. So with these three papers everything is covered. And here you have the references which are more on each chapter. If you are interested in information theory, the best book is this one, like cover and soma elements of information theory. This is a great book with a lot of examples and exercises and very intuitive. So here is the Maxwell Demons and so on. And then here you have the original papers, let's say, of each topic. So if you want to go to the original papers. And then you have the exercises. I guess you are doing the exercises but Leah and I didn't get any, well, we got some questions. But we have, I mean, you can ask, I'm usually in the afternoon here and Leah is also here. And you can write by email or in the coffee break or whatever you like. But please try the exercises. And I think on Thursday evening or afternoon, Friday, Leah will, you will have a discussion session but you should try the exercises before Leah or any one of you solve the exercises in the blackboard. Especially there are a lot of exercises based on the serial engine. Remember the serial engine is a box where you put the piston, you measure and you move the piston all the way in an expansion. And it's very interesting to see what happens if you have error in the measurement. Error in the measurement. Before, another thing that is fun maybe is that if you want to play, this is Leah and I were in a project with Natalia R.S. in Oxford on Nanotubes and we were trying to, we were trying to interpret nanomachines as Maxwell Demons and one of the people in the group, Brandon, he made this, this game. You can think what Maxwell Demons means. And it's not easy. You have, you can open the door but it's really hard to open. No, I did it. I was able to put all the, no, all the blues are very slow so you have to think very long to move it. But I managed this morning to put all the reds in one side except one, except one. So you can play with it. And, okay, and before going on a couple of notes on yesterday, yesterday was just a brief account of information theory, very, an introduction. First, I mistake that Elgar notices that the, I said Steiner Lemma for this probability of error. Remember, I think it's guess Q, the real is P, no, or something like that? What was guess P if the Q is the real? And I said that this is asymptotically 2 to minus N dvq. And this is called Steiner Lemma, right, at the ER without, so it's Steiner Lemma. And you can see the rigorous formulation and everything in Covert's book if somebody is interested. But this is a very interesting application. This is why the Kuhlbach library version is so important. Yeah, of course, these two comes, yeah, sorry. This is something that you have to get used to it when we write all the things that we have written yesterday like minus P log P and so on. And here we wrote this as P log P divided by Q. You can, as I said, you can measure all these quantities, mutual information, entropy, relative entropy. You can measure in bits if it's log of base 2. You can measure in nuts if the log is the natural log or you can measure in joules per Kelvin if you multiply by the Wolfgang constant. Steiner Lemma in information theory they use bits. So this is the result in information theory and this is, and you see here these two comes because you can put this, so here we should specify the, otherwise it would be like that. And Edgar has a very nice work on how this thing, the Kuhlbach library entropy is related with, when you apply this to P is the probability of something in the forward process in the process. And Q is the same probability in the backward process the Kuhlbach library is related with the entropy production. It's a measure of irreversibility. This is something that we have worked on for many years that you can use. The Kuhlbach library is a measure of distinguishability. So if you apply it to forward and backward processes it's a measure of irreversibility. It's a measure of the arrow of time. So when this was the first time we were able to quantify the arrow of time. And there is a lot of interesting literature on that. And now people are interested in measuring irreversibility in complex systems. But we don't know why. Okay. And the other note, the other note is something that I discussed with Matteo that entropies define like that, no? The entropy of a discrete random variable is like that is minus Px log Px. And this is a discrete sum. So this is a number between 0 and 1 and dimensionless. And log of dimensionless is dimensionless. So everything is fine here. When you have a continuous variable you have to replace the you have to replace the sum by an integral. And the P's are no longer probabilities are densities. This is a density. And the density has units. This is the units of this, the units of, for instance, if x is the energy of a particle the probability density has units of energy minus 1. And if x is a position it's position minus 1. So this has dimensions. And this is the problem because here if I change dimensions, I mean changing units, I mean if I change units, sorry, if I express this in different units there is a, here this is x minus 1, this is x, so this is 5, this is dimensionless. But the log of rho, if I change the units, if I express rho in joules or rho in electron volts, I have a factor. And this factor is an additive factor in the entropy. So this is not so well defined. This is called, this is usually called differential entropy. And also because P here is a probability. So this is between 0 and 1. But rho is, can be anything. Because it has units. So it, you can multiply by whatever you like. So the only requirement for rho is that it is positive. So this could be even be negative, this thing, this differential entropy. And you don't have the, remember what was the interpretation of this. This is the number of bits to describe x. And in differential entropy there is also the same, there is something, some interpretation which is also, this is the number of bits, well, by the, this is the number of bits, this is log of 2. This is the number of bits needed to describe x up to a precision 2 to minus n. Okay, sorry. This is the number of bits needed to describe, if you sum n. So this is a, sorry, h, h plus n is the number of bits to describe x up to a precision 2 minus a. So if you want, if you want to describe x as a, instead of decimal expression, binary expression, you will have some digits. If you want n digits, this means a precision 2 to minus n. You want n digits in your description. Then you need h plus n. So this is, but things are much easier for discrete variables. For continuous variables you have all these technical problems and so on. So these are two notes for what, the lesson that we had yesterday. Any questions? Yeah, this is okay. This is a topic of another course, let's say, but one can prove the following. Maybe we will prove it. You have any, x is anything. Any, any, so you have a process. It cannot cycle or whatever or machine, or even in the stationary regime. And you pick an observable. This x is anything, one observable. So, and you measure this in the forward, you run the process forward and backward. And then this is the forward, and this is the backward, which is more or less maybe like that. You pick a time, and this is, and you pick an observable. And this is the forward and the backward. You make a histogram of, you run the experiment many times. You make a histogram. And this is, k times this is bigger. It's smaller than, sorry, the entropy production. Yeah, this is smaller than the entropy production in the forward process. Yeah, if the process is reversible, then this is zero and there is an entropy. This is like that, no? This is zero. Okay. Sometimes this is equal. Sometimes this is less than and sometimes this is equal. And we know when, so if you observe reversibility in a process, maybe you observe reversibility, but there is some, some hidden entropy production there. In equilibrium, of course, this is zero and this is zero and this is zero. No, no, this is for any time. So, x could be any observable at any time. If x, actually we know that if x is the thought, it can be even a functional observable. Not an observable that depends on time, but then depends on the process. If x is the work, this is an equality, which is related with Krug's theorem and so on. But okay, this is no, this is something else. People are interested, so I'm answering the question, but this is beyond the course. N is just a number. So, if you want to, if you want to describe x with n binary digits, this is your precision, then you need h plus n. So, well, h plus n could be less than n. So, it doesn't mean, when we say, we say in average, how many bits you need in average to describe something. So, if it is uniform, it's clear that the number of bits is n. Because if it is a uniform, for instance, if x is a number between zero and one in a uniform distribution, you need to describe all the digits. So, h is zero in this case. Yeah, because you have log of zero, no? Suppose that you have a number between zero and one, and rho x is uniform. It's one. Well, it's one if x is between zero and one, and zero elsewhere. Okay, h in this case is zero, because it's the log of one. And then how many digits do you need to describe this number, which is a random number here, with a precision of n binary digits, n binary digits, of course. But suppose that this is not uniform. Suppose that this is a, this is something like that. Then this, these numbers are more likely. So, you can play with that. And then, and then make, describe this number x with less digits. Probably this has less entropy. And negative entropy actually. And it depends also on the, on the, on the units. Because also the, the number depends on the units. If you use other units, then the number is different. So this is just to, this is just to, to point out that information theory for continuous variables is more boring, because there are a lot of technical issues.  By the way, something that I didn't say yesterday, but I think it's important because this was, for me when I started to, to, to learn information theory, it was kind of disappointed because, but I was maybe 18 or something like that. So, and because I thought it was a theory, a mathematical theory of, of, of how we think or how, or the language or how we express thoughts and things like that. And then you say, you see that it is completely, it's just, it's just a theory of random variables. And, but this was the great, and this is in the first paragraph of, of, of Shannon papers. It says, we start this theory by neglecting semantics, by neglecting the meaning of the messages. So we want to, we want to, to make a theoretical, a mathematical theory of messages, of communication. But to do that, we have to neglect the meaning and think only on the capacity of, of some string of bits to transmit information, but not on the content of that information. This is why a completely random sequence has more information than, I don't know, a text like, to be or not to be or that's the question or something like that. So it is, for us, to be or not to be, that's the question has more information than just random numbers. But for Shannon or for the theory of information, it, that's because, because in the natural language, you have a lot of redundancies and a lot of, so it has more information from the point of view of the information theory, a completely random thing. Because information theory doesn't care about the meaning. Okay, so that's enough for, that's enough for information. Now we go to thermodynamics, I think. Here I don't have, well, and here you have also, here you have also this, the, in, in, in today's, oops, ah, this is, this is not my computer. So in today's, in yesterday's slides here, Tuesday, slides, here you have, ah, nodes, this is not, ah, these are handwritten nodes, and these are the nodes for lesson two and lesson three. So it's on, on the web page of Thursday, but it is, ah, okay. But the nodes are the nodes. I mean the, the, if you want something more, ah, rigorous, let's say, you go to the papers. Okay. So, ah, thermodynamics. We are going to study some, ah, properties of thermodynamics, ah, but we are going to focus only, ah, in, in systems that are in contact with the thermal bath. So you have a thermal bath at a given temperature, and you have also some external, ah, the, the, ah, the energy exchange between the thermal bath and the system is what we call heat. In these situations where you have a single thermal bath and an external agent, everything is very clear, as I was going to show you now. Ah, and, and, and you have an external agent, and the, the energy exchange between the external energy and the, and the system is the work. Ah, so in, we are going to study first this heat and work, and give some expressions. First for, for general systems, and, and then we can, ah, we will, we will make a particular case of discrete systems, which are also much simpler. So, ah, you have the, ah, we, I will call X the, the, ah, the microscopic state of the system. Although we are dealing with, ah, ah, classical systems, everything is not so difficult to generalize most of the results to quantum systems. And, ah, so this is, I prefer X, but, and X could be even one of these easy models that you are studying with this act, that you have a plus one minus one. These models are, you can have discrete classical models as well. Eh, if you have, ah, a classical system, that in, for, for instance of, is, instead of spins, you can have a bistable system that can be only stable into, ah, in a double world potential. And, and this is, so don't think that discreteness means quantum is, you can have also classical systems, ah, with discrete states. So, you have a microscopic state here. And, ah, you have an external agent that, ah, ah, ah, can manipulate some parameter of the Hamiltonian. So, this is the parameter. You have a Hamiltonian, and the Hamiltonian depends on X, and depends on the parameter, and because of the external agent, this, ah, Hamiltonian is time dependent. So, this is the Hamiltonian of my system. And then I will assume as well that you, we have, ah, probability distribution over the micro, of the microscopic states of the system. So, this is, we, we can call this the probabilistic state of the system. Oh, it's done. No, lambda is no time, lambda is, ah, T is, ah, is time, yeah. Yeah, you change, the idea is that you have a system, and you change something, think of the Carnot cycle where you change the piston, change, think of the Cillar engine where you change the piston like that, or think of, ah, a field that you can switch on, switch off, or modulate, or whatever. So, lambda could be anything, ah. In fact, ah, ah, ah, for instance, ah, he used this lambda, adding this field, no, lambda by times observable, this morning, I saw that you are doing. This is a, this is a particular case, where you can have any. So, ah, this probabilistic state, you have different, ah, different theories, or different, ah, yeah, theories that gives, that tells you how this evolves in time. Maybe some of you know the Fokker Plan equation, for instance. If you have a system in contact with a thermal bath, if you have a discrete system, this could be, ah, this could be, ah, master equation. If it is, ah, if it is a quantum system, you can even, this could be a density matrix, and you will have something called the Limblat equation, that you probably do know. So, the, we have an open system. So, an open system is all, is, is, is in contact with a thermal bath, will have a kind of, ah, stochastic evolution, and this stochastic evolution is, ah, ruled by one of these equations. So, this, this will have the, we will assume that this, well, that we can, in principle, know this evolution. This could be, you will, ah, if the, well, if the system is isolated. So, ah, let's put Fokker Planck, ah, master equation, Limblat, et cetera. So, ah, ok. So, ah, how can we, ah, ah, calculate heat and work here? If you compute, ah, first, let's compute the energy. I will call this the energy, because the Hamiltonian depends on time. The energy is not a constant of motion, and the energy is actually, ah, this, ah, this average over x. So, ah, I have to, ah, integrate h. And now if I, if I calculate how the energy changes in time, the derivative, ah, I can put the derivative inside, and I have two terms. The first term is, ah, the derivative, the partial derivative of h with respect to lambda times lambda dot. Let me put it like that. And the second term is, ah, the Hamiltonian and then the derivative of rho with respect to tau. So, you have, ah, that the, the energy changes because of two things. One is, ah, the modification of the, the action of the external agent, and the other is the evolution of the, of the, of the rho itself. So, this is, ah, one can check that this is heat, this is work, sorry. This is actually the exchange of energy that the external agent introduces. I mean, the, the energy that the external energy introduces into the system, and this is, this is heat. Can we, ah, prove this? Well, ah, the proof, ah, the best way, for me the best way to, to understand this is, is to consider a quench. A quench is, is just, if you have lambda, is to consider that lambda is constant, and at some time, ah, you change the value of lambda to some other value. In this case, what is the work? The work is the energy that you introduce in, since, since the, the, this, this change is instantaneous, rho doesn't change, and, and there is no time for the system to exchange energy with the heat bath. So, the, the, the increment of energy due to this change is work. Okay? And this is essentially this term here. After this happens, the system here, suppose that the system here is, is, is in equilibrium or in a stationary state, and it's happy here, but then suddenly you change lambda. Then here you have a relaxation. And in this relaxation, the system is losing or absorbing energy. And this is heat, because the, the external agent is now, the external agent only operated here. So here the external agent has no action. So you see here the work is concentrated in the quench time, and what the, all the exchange of energy in the relaxation is, is heat. And if you apply this formula, you will see that here, what is happening in the relaxation is that the system is encoded with a thermal bath and rho is changing. And the, the Hamiltonian is constant. And here is the other way around. You consider rho constant. And you see the, the, the contribution to the energy of the system due to the, to the action of the agent. Okay? So, and this is the first, this is my unit of time. This is everything per unit of time. So I, we can write this like that. And this is the first law of thermodynamics. And of course you can integrate in, in, in a process and so on. And also to be con, to convince that this is a work and heat. It's also interesting to consider a quasi-static process. In a quasi-static process, lambda is, is, is very small. Let's put it like that, lambda is small. I want to put lambda close to zero because lambda has units. My students, I always say, if you have, if you have something has units you cannot say lambda much smaller than one or something like that. No. Let's say lambda is small. And what happens in a quasi-static, in a quasi-static limit if a system is in contact with a thermal bath? Anybody know what is the consequence or what is, it is in a clear room at any stage of the system, of this, of the process. So if, if, if I have a quasi-static process, rho x t, rho x t is very complicated. As I said, you need to solve the Fokker-Pann equation. You need to solve the Limba equation or the master equation and so on. But you know something that the steady state is the equilibrium. So, and the equilibrium is the canonical.  This is, the system reaches this, this state. No? And if the, if the process is very, very, very slow, then the system can be in equilibrium all the time. Of course if the system is fast like the quench, the quench is the opposite. Here you are in equilibrium and suddenly you depart from equilibrium. But if the process is small, if this is small, you can have this. And then, and then this is a, let's call it like that, rho q, rho aq x lambda. This is Gibbs state, yeah? The thermal state or whatever you like. Yeah, you can have, this is for systems in contact with a single thermal bath. Of course if you have different thermal baths at different temperatures, then things are more complicated. But I want to just to, to look at this example. And you can also have other, what is called generalized Gibbs states, which are the exponential of minus beta h plus some other quantities. Conserved quantities, yeah. If you have more than one conserved quantities, this is, thermal states are based on the, on the assumption that the only constant of motion is the energy. Well, actually, when you, when you write down the Fokker-Panne equation, you force the equation to reproduce the Boltzmann distribution for a long time. And this is the fluctuation dissipation relationship. So if you are describing a system, you have to be sure that your theory, your equation, has as a steady state the thermal equilibrium. And this is fluctuation dissipation. In the case of Langevin equations, you have to force this. In the case of Maser equations, you have to impose detailed balance, which ensures you that the steady state is the equilibrium. In the Limblad equation, you also have to, detailed balance to put, detailed balance. And if your equation does not reproduce the thermal state as in the long limit, then it's wrong. Then it's wrong. Then you can make a perpetuum mobility of the second kind. And then you, so you, this is a requirement of any theory that you have for describing a system, a physical system, is that if there is only a single thermal bar in the limit of T large, the steady state should be thermal. Well, I call the steady state of the Fokker-Panne equation. And the steady state of the Fokker-Panne equation must be the thermal, the equilibrium state. Yeah, equilibrium, I mean equilibrium is a, it's a particular case of steady state. Usually we say non-equilibrium steady state, when we want to say that the steady state is non-equilibrium, we use this non-equilibrium steady state. Okay, so if you put this here in the work, we put now this, and then here we have, well, this goes out of the integral, so you can put, this is the work. And using this trick that, I guess, Isaac also used it to prove that the average of the, of this observable is the partial derivative of log z. You can take this out of the integral. This is for a given time. I could even remove the time, but it is just to keep the coherence with this expression. Now you have this, the derivative of, and the exponential, and you can write this as the derivative of the exponential. No, this is, I make the derivative, so I have to divide by beta and put a minus sign. No, if I, if you make this derivative, I get the exponential again, evaluate it in lambda t, and then the, and then minus beta, which cancels with this minus beta, times the derivative of h with respect to lambda, evaluate it in lambda t, okay? And now you can get this out of the, we can write this as kt minus lambda dot z. And beta can be kt, beta is whatever kt, and this is, we can take this out of the integral, and I recover the partition function, the definition of the partition function, so this is the partition function. So the work, the work, this is the power actually, the work per unit of time. This is minus kt, minus lambda, and here is the derivative with respect to lambda of a, let me write this minus here, of minus kt log, in this case it's natural log, and this is what we call free energy. This is the free energy in statistical mechanics. So you have that the work is lambda dot and the derivative of the free energy with respect to lambda, okay? And now if you integrate over a process, let's integrate this over a process, wt w, which is between some t0 and t final, or initial and final, of d omega t, sorry, omega, sorry, no, dt wt, you put this here and you see that this is an exact differential, this is the derivative of f with respect to lambda, the derivative of lambda with respect to t, this is the anti-derivative of this, so this is f, so this is f at the final minus f at the beginning. So this is delta f. So the work is equal to delta f in a quasi-static and isothermal, because the system, because this is quasi-static, the system is always at temperature t, so quasi-static and isothermal process. And this is a result of thermodynamic that maybe you remember. W is not an exact differential, so you cannot put w as the difference between two, between something at the end and something at the beginning, it depends on the path, but if the process is isothermal, then the work is just a difference of free energy, okay? And this is in quasi-static, and the second law if you like is that w, this is second law, that one can prove from here as well. The second law can be proved using Fokker-Pann equations or one of these equations, and one can prove that the work is always bigger than delta f, and it's only equal to delta f in equilibrium. So work is always the work that the external agent is putting into the system, so the extracted work is minus, and then this is telling you also that delta f is the maximum work that you can extract in a process when delta f is negative. So when you go to something with high free energy to low free energy, you can extract this difference as work. And if you do this reversibly in the quasi-static limit, then you extract this amount of energy. So this is why it's called free energy because it's the energy that you convert into work. But always isothermally. If it is not isothermal, things are more complicated. But this is just an illustration, yeah? Very good question. This depends on the time scales. What do you think? In an expansion of a gas, what do you think is the speed to reach this in an expansion? What do you think? How fast you have to move this piston? Yeah, it is related with the... It is related with the relaxation time of the system. Every system described by a Fokker-Panne equation has a relaxation time. Usually it's related with the eigenvalue. You know a Fokker-Panne equation has an eigenvalue equal to zero, which the eigenvector is the stationary probability, and the next one tells you how fast the system relaxes. So the time lambda must be much slower than this eigenvalue. But this will be important because in information devices, there are huge time scale separations. For instance, when you have a double-well potential, and this is interesting because we will discuss this tomorrow, when we have a single-well potential, a double-well potential, you have a time scale for the relaxation within a potential, which is maybe it could be depending on the particle, but it could be milliseconds or seconds, let's say, if it's a Brownian particle and so on. And then you have the time scale to jump, and the jump is necessary for a global relaxation. And this could be of the order of millennials or the age of the universe. So you have too huge difference in time scale. When I say lambda is small, it's clear that there is no this type of strange things. And that you have... And lambda must be much slower than the relaxation processes in the system. And this is what allows you to be all the time in equilibrium. We need the Gibbs state, yeah? No, no, from a Gibbs state, if you go from a Gibbs state with high free energy to a Gibbs state with low free energy, of course, you can't get work. No, you cannot extract work in unitary... This is in quantum mechanics using unitary evolution. But using thermal baths, you can extract work. But using unitary evolution, you cannot... Yeah, it is true. If it is isolated, if the system is isolated, then you cannot extract work from this state. This is called passive states. And it's because you cannot extract work in a Hamiltonian evolution. But if you are in contact with thermal baths, you can extract work. Yeah, suppose... I mean, the simplest case is suppose a particle in a potential like that. If you go from this to this potential, you extract work because the particle... Unless the particle is just in the vertex, in the minimum, if the particle is here, it has some energy. And let's do like that. So you have your particle, for instance, here. And you change the intensity of your... This is an optical trap, so you change the intensity to this one. So the particle loses this energy. And this is the work. Another thing is that in an optical trap, how do you use this work? I mean, nobody knows how to extract this work. But in principle, this is work. And the same. This is like an expansion. We have also with Edgar a Brownian cycle where instead of a gas, we use a Brownian particle. And the expansion is like that. And the compression is opening the potential. And the expansion is closing the potential. And we did this with optical traps in the laboratory. So yeah, passive states, you cannot take energy from this state using unitary evolution or in quantum mechanics. Yeah, some people call it like that, but I don't... I'm not familiar with... I don't know. Yeah, but this is when you have... I think in the aerotropies, when you have different thermal basins like that. Then I don't know. Okay. So free energy is very important. And one of the things that we want to do is to apply this to all what we talk the first day, the Maxwell-Demon, the Cedar engine, and so on. Processes where information is important. And processes where information is important, the main thing is that they are out of equilibrium. Why? And this is important because information thermodynamics is actually statistical mechanics but applied to things that can store information or can manipulate information. And what is information? Maybe this reflection is for the end of the course, but it is good to advance this now. And if you think of a hard drive or DNA, which is also a system that stores information, what do you think is the most important characteristic of those devices? Reliable? And what means reliable in terms of physical properties? Student change. Student change. So you need something that has long life states. But equilibrium is long life. equilibrium is eternal. But you need something else. What is this something else? You need states with long life, but you need more than one state. Because you need the 01 or GTAC in the case of... So you need systems that have different states. They have to... Those states must be long life states. And what else? I would add something else. Because, okay, you have 01 in a hard drive, GTAC in a DNA, but these states can be anything, or you need to manipulate the zeros and ones in a bit. You need also that they are... that you can go from one to the other. Maybe... They need to be stable, but by some mechanism, you need also to go from one to the other. And this happens in DNA mutations. But also, in DNA, you don't... I mean, mutations are not so good. But in DNA, you also need something else. That all the machinery, the chemical machinery that manipulates DNA for replication, transcription, and so on, must be such that the four bases, the GACT, must be similar in which respect to the machinery. Because otherwise, you would need a machinery for G, a machinery for E. And the same thing works with the four. So they must have some similarities. And the same for zeros and ones, because it is just the states of magnetic domains. So you need distinct states with a long life, but at the same time that manipulating some parameter, you could jump from one to the other, and that they must be somehow equivalent, or they must be similar. And the minimal model for that is the double-wit potential. Actually, if you think of this idea, what you need is really a symmetry breaking. You need a system with a symmetry breaking in which you know what is a spontaneous symmetry breaking. You know when your Hamiltonian is symmetric, but the system breaks the symmetry, and it goes just like that. Here, the system, the Hamiltonian is... Here, the Hamiltonian is this potential. This is the potential. And the potential is symmetric. But if you have a single particle, if you have a bunch of particles, you will see this density. But if you have just one, or if the barrier is very large and you start with particles on the left, they will remain on the left. So this system does not equilibrate. This system breaks the symmetry. It's not the symmetry breaking that you are used to, which is based on many degrees of freedom, on the easy model. But the idea is that you need this... this kind of symmetry breaking. And this is, at least for me, the main characteristic of information devices. So to summarize what is now thermodynamic of information, thermodynamic of information is thermodynamic applied to this type of Hamiltonian, this type of devices. And in these devices, you have... that you never equilibrate completely the system. You never reach a global equilibrium. You reach the state, even though you do a quasi-static process, but quasi-static in the sense of relaxation within each well, but not so slow, because maybe the time to jump here is the age of the universe. So you can never reach this stationarity. Now we have two hugely separated time scales. One here, the evolution here, and the other is the jumps. And if your process is in between, it's quasi-static with respect to each well, but it's not able to equilibrate the system globally. So this is what happens, for instance, in this sequence. This is an example. We have a barrier like that, and if you lower these, you know that if you apply the equilibrium, I mean, if this is really in equilibrium, particles will have more probability to be here, because the energy is lower here than here. But of course the system cannot... If you come from here, the system cannot relax. It will take the age of the universe to relax, because you need the jumps. So the state will remain like that. One half, one half. If you come from here, think that you come from here, you lower this well, and then this is a state which is out of equilibrium. It's out of equilibrium. Because it's not the state. In the Gibbs state, this peak should be larger. It's local equilibrium, but not global equilibrium. And this is one of the features of information devices. So you have non-equilibrium states like this one. This is a non-equilibrium state. So all these things are not valid, so we have to modify these things. And thermodynamics is... The thermodynamic information is just a theory of this type of state. And this type of... Even the state can depend on the information that we have about the system. If we now measure, if it's a single particle, and we measure and the particle is on the left, then suddenly we have a kind of collapse of the probability density, and this is again non-equilibrium. So you see that the state of the system of these devices depends on the history, because it depends on where this comes from, depends on the measurements that we do, depends on the information. They are mostly in equilibrium, because we are locally in equilibrium, but the global state depends on a lot of things. It depends on the information that we have. It depends on whether we have measured or not, et cetera. And we would like to have some theory like this one for this type of states. And this is what we have is... This is provided by something called the non-equilibrium free energy. And the non-equilibrium free energy, which is only valid for systems in contact with thermal baths. So we still have this scheme, the thermal bath. We can generalize this, but let's focus on this scheme, because this scheme is the Schiller engine as well. And the work done by the external agent. But now we don't have this identity. We don't have this identity. Actually, we don't have even... The system is no longer the Gibbs state, so we don't have a partition function. So we don't have even a free energy. But even though we don't have all these things, we can still define a free energy for non-equilibrium states. And this free energy is... We use this letter. This is the non-equilibrium. And it's defined for a rho and for a Hamiltonian. So it's a function. It's a number that you give me a probabilistic state of the system like this one. And you give me a Hamiltonian like this one. Well, here we don't have kinetic energy. The kinetic energy in these examples of wells and Brownian particles does not play any role because it's always in equilibrium. And I can calculate the free energy. What is a free energy? Well, you remember free energy is in thermodynamics is E minus T s. So this is the most... This is going to be the energy minus T and the Shannon entropy. So this is a minus K sum over X of rho X log rho X. And this is the non-equilibrium free energy. Yeah, some properties. If rho is the equilibrium one, then this is F. This is minus KT log C. So the non-equilibrium free energy applied to equilibrium states is equal. Another property which is nice is that the... If the system is not in equilibrium, the non-equilibrium free energy is equal to the equilibrium free energy plus KT the cool back library distance between the rho and the rho equilibrium. And this is always positive. So this means that this is always bigger or equal than the equilibrium. So the non-equilibrium free energy is always bigger than the equilibrium free energy. And the final property or the most important property is that what it's going to allow us to use non-equilibrium free energy to obtain the energetics of processes where information is important is this... This is not the property. This is just the main result is that in a process the work needed to complete the process is always bigger than the non-equilibrium free energy the increment of non-equilibrium free energy. So the non-equilibrium free energy has the same meaning than the equilibrium free energy for non-equilibrium states. It tells you how much work you can extract if you go from up to down or it tells you how much work do you need to put in a system to go from low free energy to high free energy. From low free energy to high free energy delta F is positive so you need to put work. Remember that work is always positive if you put work into the system. In a motor when you extract work W is negative and delta F must be, of course, negative as well so you need to go from high to low. No, the particular function is only defined for the Gibbs states. I mean if you have one of these generalized Gibbs states, you mean or what? Yeah, yeah, yeah, go on. For non-equilibrium states we can consider in the existence of some operator, for example H0 and the non-equilibrium states can be equal to the exponential of minus beta H0 at Z0 where Z0 is the truss. Yeah. But here H0 is not the aminitronium of the system it's just operator for defining some new states, some non-equilibrium new states. Yeah, he's telling that and we can have in principle you can write any row as exponential of minus something beta divided by something. Yeah, but this is this has, in principle, it has nothing to do with the non-equilibrium free energy. I mean maybe for some, maybe you can do this. Yeah, if you have here something A and here you have N of course log of rho minus log of rho is A minus log of N. So you can play with that but maybe you can find some expression for the non-equilibrium free energy. Yeah, I don't know. I'm not sure but... I see that because for some states for some non-equilibrium states the partition function the truss at the exponential of minus beta H0 for example is equal to the partition function of the... Well, let's discuss this later because it is a very technical thing and it is nothing... Okay, so maybe you want to prove of that or not, are you happy? This is the main resource of statistics of... This is the main tool I mean just applying this you can explain the CLA engine everything. This is the thing that we are going to use on Friday and to understand the CLA engine and so on. Yeah. This is also... This is an interesting question that we have... This non-equilibrium free energy is I think the first time it appeared in the literature was in the 70s. In a very strange book that I never... I never found it. It is something like the drops that come out from a faucet or something like that. This is the name of the book but I never found it. But it was cited as the first time this appeared, this notion appeared. And then we have used it in the 90s and in 2000 we have used it for not for this for thermodynamic information but people have used it for a lot of different things. In stochastic thermodynamics it is well used. And I don't know for active systems in principle you can use it as well. Although the problem with active... Yeah, for active systems you can use it as well. But tomorrow very slow because today the plan was to finish this lesson but I will finish tomorrow. But tomorrow we will see molecular motors and we will see that in case of... most active systems what they have is a consumption of some fuel like ATP or something like that. So in this case the problem is more complicated. You can do this as well you can do this as well. But it is more related with the production in the environment in this case in the fuel. So tomorrow maybe we can discuss on that. Is there an equivalent of a quasi-static process where you get the... Okay, this is in a process and there is the... Okay, I'm glad to make this question because I... Nobody will ask for the proof but the proof of this is actually... Oops Actually the proof of this is in... The proof of this is spread in different papers in the review that I showed you I think I think there is a very rigorous proof in one of the reviews. Yeah, there are... And also van den Broek is supposed to have a proof and I don't know how to do this. Okay but I will show you not the proof rigorous proof but but this is in our review in nature this picture and it's just an idea of why this is so and why this... and the question to derive this we are thinking of a process now it could be quasi-static or not, let's discuss this in a moment, but a process that connects something which is out of equilibrium because delta f is the final non-equilibrium free energy minus the initial equilibrium free energy so I want the process that the initial state is non-equilibrium the final state is non-equilibrium well to make it easier let's discuss the case where the initial state is non-equilibrium and the final state is equilibrium because it's easier so the idea that you have we represent the Hamiltonian in this case it's a potential like that and this is the state and this is clearly out of equilibrium because exponential of the potential is not equilibrium so if you let a relaxation if you let the system relax this is equilibrium this is more or less the Boltzmann state corresponding to this potential or the Gibbs state corresponding to this potential if you do this what is the work and what is the heat you just let the system relax without doing anything heat work is work is zero and there is the dissipation of heat or the contrary maybe sometimes the system adopts the system as some so if you do that you don't get any work but what is the best thing you can do if you want to extract work from this from the state so supposedly this has a some capacity to provide energy to you okay it's easy to see that the first the best way is the following is to modify the potential in such a way that the state is in equilibrium and you do this instantaneously you go here instantaneously it's a kind of cheap trick because it is I mean you start out of equilibrium with the system in equilibrium so this is instantaneous and then for now you are in the green part of the plot which is equilibrium so now you can use equilibrium thermodynamics you can use what we have derived at the beginning of the class so you can statically go from here to here and then you have and then you go there if you do that you have to prove that I mean it's so easy that I can prove it here if I have five minutes that what is the work the work is there are two steps one is the instantaneous quench if you like quench means that that it's very fast something so you have a quench and the quench is that this is this is instantaneous so there is no heat and the only work comes from the difference of energy from here to here so I have this is H0 this is Hamiltonian and this is H rho H rho by the way if I want H rho must be such that rho is my initial state by the way this is a nasty notation because rho is the initial and rho 0 is the final and it's because it's in equilibrium with H0 so you have such that rho initial is the equilibrium so this means that rho initial must be exponential of minus beta H rho so this means that H rho must be kT log rho initial so this must be maybe it's impossible but we are trying just to make a general statement so let's say we start with a rho the idea is first instantaneously we change our Hamiltonian from H0 to kT log rho and with this trick this is a new Hamiltonian let's call H what is the notation here H rho and with this trick let's actually go to equilibrium and now quasi statically we go from H rho to H0 again that's it so what is the work in the first quench is I've changed from H rho to H0 initially I have an energy H0 average over rho and finally I have an energy H rho over rho and then in the quasi static which is this one from H rho to H0 everything is in equilibrium so I can apply what I have derived before in the previous the free energy of 0 and the free energy of rho the equilibrium free energy okay so now if you sum the total work if you sum these things we have F0 F0 is H 0 in rho 0 because it's equilibrium and minus T the Shannon entropy of rho 0 F rho is H rho in rho average over rho plus T S rho and then I have to sum this which is H rho minus H0 in rho and things should sorry it was fast and things should cancel this cancels with this and then I get H0 in rho TS rho this is the final free energy this is the final sorry this is the initial free energy this and this is the initial free energy non-equilibrium and now it's non-equilibrium because now you see I'm averaging H0 over rho which is not the equilibrium distribution this is the important part and this this is the equilibrium free energy so this is the final this is the final because of this rho this is the initial rho so this is the initial sorry that it was a bit fast but if somebody has problems it's a one line calculation so it doesn't and then two more things we have derived this then well first thing when this is equal we have derived that this is equal but I've shown you that there is a process where I can go from here to here extract well making this minimal word delta f or extract the word if you like okay and and this process is reversible and you say my god it's reversible but if everything is non-equilibrium well if you think it's operationally reversible in the sense that if you do this thing in the you run the protocol the protocol is a quasi static change from this potential to this potential and a rapid quench from this potential to this potential the process is indistinguishable the forward and the backward even though the systems are in non-equilibrium so this is also important the equality is rich when the process is operationally reversible which means that it is indistinguishable the state of the system the process and in the backward process and to end and you can say oh my god so if everything that works in equilibrium now works out of equilibrium that's great no but of course here there is something which is very strange which is the quench I mean I mean it's like cheating you start out of equilibrium but immediately you are in equilibrium so I mean who cares about that it's like you are in equilibrium just during a millisecond and of course of course this is nothing but the nice point here so this means that this couldn't be applied to a real system I mean you are in initial state it's just let's say I mean because it uses that initial state but this is very important for information devices because in information devices you have the huge the huge separation of time scale so you can have non-equilibrium states which have a very long lifetime which are essentially stationary like the ones that we have seen in the two worlds so then we can apply that because then this can be non-equilibrium in the sense of two worlds and maybe and it's a state that is completely stable but out of equilibrium and then you can do this process and so this scenario although it seems very artificial it's something that you can do in information devices as well and this I don't know if this answers your question but it is more honest and that's it so sorry no time for questions if you have questions I will be around and we can discuss whatever and do the exercises and Liya will be also around this afternoon I have to


Hi everybody. Can you hear me? All right, fantastic. My name is Darius. I'm a Mozilla fellow and I'm working with Code for Science and Society on issues of the decentralized web. And I'm here to talk to you about a vision that I have for the future of social network sites and internet services as they might exist outside of the context of corporate monopolies. So, first off, just to define some terms, the decentralized web. I'm going to talk about it a lot here and this is kind of a classic diagram that I have added red circles to. And just very briefly, a centralized network is where there's one source of truth at the middle. Everybody connects to it and it handles all of the messaging and making sure that people, you know, if I'm on the bottom of that blob and I want to talk to someone on the top of that blob, I have to go straight through that big red circle and everybody has to do that. And I made that circle big and red because that circle is more powerful than the other circles. If that circle is like a Facebook server, Facebook can say, no, you're not able to talk to this person because I have decided that you are not able to talk to that person. A decentralized network has many different nodes of roughly equal power and then a whole bunch more nodes that are just regular users. So, this is what networks like Mastodon are like or if you look at the internet in general, you know, one of those red things might be github.com and one of those red things might be Amazon.com and that sort of thing. So in the context of the whole internet, these things are decentralized, but in the context of like Facebook, Facebook as a network is centralized, you have to go through that central control point. And then a truly distributed system is one where every node is kind of its own server and client and they can all talk to each other as equals. But I'm actually talking about the middle case here, not distributed networks and not centralized networks. I'm interested in decentralized networks specifically. The internet's been decentralized since the very start. It was initially proposed during the Cold War by defense think tanks as a resilient network that could pass information around the country even in the case of a nuclear conflict. So it's grim, but the idea is that if one city is wiped off the map, communication traffic could reroute around the big smoking hole in the ground. It was like a Cold War fever dream type thing. Decentralization might also be explained by what it is not. Facebook is sometimes referred to as a walled garden. It's a place that has cool stuff that people might want. But in order to get in there, you have to go through this sort of choke point, usually a gate, there's probably a lock on it. There's usually some form of surveillance like ground keep groundskeepers and that sort of thing. And you know, once you're inside there, it seems like a public space, but it's actually a private space. There's this difference between the inside and the out. And that's an example of centralization.  The problem, especially in the context of social network sites, is that the world itself and our social structures are not especially centralized. Different communities have all sorts of different kinds of norms. For example, we can take the example of on Facebook, there was a time when they decided to install a real name policy, they wanted everybody to be real name, they wanted people to use their government ID names on their profiles. And there was a subset of people who thought this was like in the Facebook community, like users who thought this was a great idea for safety. And there were a bunch of people who saw this as extremely harmful to them for a number of different reasons. And I think part of the problem here is that we have a big centralized social network. And you can't have, I don't think you can have norms that work for a billion people. I think communities need to be partitioned out, and they can kind of come up with their own norms. And there's nothing to say that a Facebook couldn't do that. Like maybe they could give every, you know, Facebook group admin more power or that sort of thing. But it seems highly unlikely for a number of basically organizational reasons. So my vision for a decentralized web is where it's almost kind of like everybody gets their own little wall gardens, kind of, but not really. And I was having trouble articulating this until I read a 50 year old computer science document from the birth of the Internet. In the late 1960s, ARPA, the research branch of the US military, now known as DARPA, funded a project called the ARPA Net. It was meant to be a realization of those Cold War dreams. And it was a joint project between university researchers and defense contractors and also some more general contractors in the sort of civilian technology space. On June 19th, 1970, a group of computer scientists at UCLA and Harvard University published a document called Request for Comments 55, a prototypical implementation of the network control program. The document describes a program that can live on the computer but lets it talk to the ARPA Net. On a modern computer, you might have called it network drivers or something like that. It's the code that lets your machine interpret signals going in and out of it from the Internet, or the ARPA Net in this case. There was no Internet. There also was no ARPA Net at this time either, basically. It's pretty dry until you get to this diagram, which I love. The authors of this paper actually refer to these blobs, literally I quote, as squishy amoeba-like objects. And I thought it was really fun because if you read the whole paper, you realize that there's not actually a hard distinction between what the input interpreter is and the input handler and the output scheduler. And whoever drew up this diagram wanted to make sure that while things like these cues in the middle that look like cylinders are discrete components, these other things are much more nebulous. If you've ever been to like an Amazon presentation or something, they'll always have the cloud, and it's like a cloud. And that's also kind of trying to get the same point across, but I like squishy amoeba-like objects way better because they're still objects. So this got me thinking about amoebas as a metaphor. The amoeba, or perhaps amoeboid, is a useful construct for me because it's amorphous, and yet it has clearly defined inside and outside. You know what is inside the amoeba and what's outside. That's basically it's defining characteristic as a creature or an object. They let things in and out of this kind of filtering system, in this membrane, and an amoeba can change shape as needed, as is needed to be convenient. Things can be allowed into the amoeba and also kicked out. And I think about this now when I think about decentralized and what people call federated networks, which I will explain in a couple of slides. So I want to just start with something that I'm familiar with here, which is this social network called Nastodon that you might be familiar with. It is kind of a Twitter clone. It's open source. You can run your own server. I run my own server. It's called FriendCamp. It's for me and like 50 people that I know. There's a few FriendCampers sitting in this audience right now. It looks like this from a user level. I just, for people's privacy, I replaced all the text with dashes. But I want to talk about what this service looks like on a network level. So from a user, it's like a website. You log into it or you have an app and you post stuff. It's like Twitter. So this is a server. A server is a piece of metal with, it's very complicated and it has electricity going through it. AKA a computer. These are people. And these people have their own computers or tablets or phones or whatever. And these tablets and phones talk to the server. Let's say this server is FriendCamp. And so there's people in the audience right now and maybe they're posting to FriendCamp from their phone or whatever. And this is centralized. This in and of itself is a centralized system. The difference is that for something like FriendCamp, it's me and 50 of my friends, which is way different than Mark Zuckerberg and a billion people who he's never met. It's like a different social contract. So okay, we have this little node here. And it's called Federated because these signals get sent out to other servers as well with their own people inside of them. And there might be more than that. And they all talk to each other and not everyone talks to everyone. But if you look at the top, you'll see that those three nodes, you know, two of them don't talk to each other, right? Might be because they don't know each other exist or it might be because one node hates the other node. And, you know, like one of them was a bad actor. And so they decided to block them or whatever. So this is like an example network here. What makes FriendCamp different and has started making me think about these different models is right now on basic mastodon. If I'm on one of these nodes, as long as I have a link there, all my messages are viewable by basically everybody else. It's all kind of public to the locally known network. On FriendCamp, we have this concept of local only posts. These are posts that never leave this piece of metal here. So they're inherently more secure if you trust me, which because you're 50 of my friends who are using this, you do trust me. At least, you know, I hope so. You know, it's a different, the trust that a friend of mine has in me is very different than the trust that I have even in a company that I think is really great. You know, it's like a different level of trust. So back to here. So this is where the amoeba part comes in. So this is something that already exists technically. And what I would like to do is start introducing concepts. You'll notice that in the middle, there's kind of a kite shape going from the upper left to the upper right to the lower left. And this is a bunch of nodes that all mutually follow each other or aware of each other's existence. And maybe we decide that we are friends with a bunch of people on this other server. So in addition to like, I can friend someone on the other side of the network individually as long as my server can route it, maybe we have servers that become friends with other servers. And so that may introduce a new concept, you know, that's not a fully public post, but maybe it posts to your neighborhood of trusted servers. It's not all your known servers because there are servers inside that blob that know about servers outside of the blob. We just don't fully trust them yet. But we could, you know, maybe in the future, we decide that we really like them and bring them in too. Now, the thing about this sort of amoeba like grabbing things process is there has to be social contracts involved here. So the reason that you might bring someone into your amoeba blob neighborhood thing is because you have sat down and talked with them with like community to community. And you both realize that you share the same values, you don't want the same kinds of, you know, you dislike the same kinds of speech on your network. And there's not, you're going to be a good fit for each other, essentially. And so I want to see ways to bring out these kind of medium spaces there because like local only is is cool. It's really useful. It's really helpful. It's great to have 50 people able to like talk to each other in like a more intimate space. But there's got to be something between that and like talking to the entire world and having your messages sent out to like 5000 servers, you know, 4,990 you have never heard of in your life. So what I'm hoping for is something like this in the future. This is not a mastodon only or even a Fediverse only type concept. But what I do think is that we need to limit the sizes of these individual nodes because otherwise it's not going to be, you know, at that point, if we have like five nodes in the whole network and they each have a billion people on them each, then it's that's just what we have right now.  And, you know, that's that's kind of just re centralizing. So I'm working on some documents. I'm hoping in a couple of months, I'll sort of publish something like a best practices guide for maintaining one of these nodes and how we might federate nodes with each other to get this notion of a neighborhood. Anyway, that's, you know, that's the 15 minutes of stuff that I wanted to talk about. And so what I really wanted to do is like open up to questions for people. So thanks for listening to this kind of intro.