Today's lecture will be, this is the seventh lecture, remember today we have two lectures and tomorrow is the last one. Today will be about close to equilibrium conditions so systems that are driven out of equilibrium, but the force that is driving the systems out of equilibrium is weak and there are very interesting results for this situation. Please remember that what I have been explaining was general, was for any force, any degree of non-equilibrium, and today I will focus on stochastic thermodynamics of systems weakly perturbed out of equilibrium or weakly perturbed out of a non-equilibrium steady state. I will also think about a system in a steady state in which we apply a tiny, tiny dependent force, right? Okay, so before starting, I have divided my lecture in two parts, the first will be a lot of math and theory and the second will be experiments about these results. Before starting I'd like to request if you have questions, if it's not a problem for you to switch on your camera this way, if it's not a problem of course, this way I can see who is making the questions and it makes this course much more human, all right? Fine, so I'll go now to share the whiteboard because I'll do most of my calculations this way. So, okay, so today lecture seven, I will start with a very well-known result in stochastic thermodynamics, which is the so-called Harada-Sasai quality. Okay, so this will be my first result I will discuss the Harada-Sasai quality. This is an equality that connects dissipation or end-reproduction with how much a system does violate the fluctuation dissipation relation. Okay, this is a very nice result. And, okay, this is found, for example, in papers by Harada and Sasa, in PRL, this was the first contribution was in PRL, Physical Revive Letters 2005. The first one is PRE in 2006. I highly recommend you to vote the PRE. To be honest, I've gone through both and I understood much better this one. However, this one was the first, which they introduced this result. And also a fantastic work also by Harada only, okay, this is also Harada and Sasa, and a fantastic work by Harada, which is in PRE in 2009. Okay. This is Takahiro Harada and Shinichi Sasai, if you want more information. All right, so I will show an equality for a group of systems, which is non-governed dynamics, but this is coming generalized to more situations and recently there has been also a generalization to field theories. Okay, so I start with a non-governed system, which is the following. The system is over-damped, there is no mass, or the mass is very small, there is friction from the bath, there is an external force, okay, a force which could be conservative or non-conservative. Then there is a perturbation, which we will call this one, epsilon, Fp. This is a time-dependent force that is applied in the system. Epsilon is a parameter that we would say it's small. And finally, there is the white noise for the semi-mass. Okay, remember, this is a zero-mean and autocorrelation, given by a two-gamma committee. Okay, I'm not being very, okay, remember, this is a delta function. All right, so what is the path probability? The probability for a trajectory in this model. The path probability, which I usually call it like this, BXT, or a trajectory, any trajectory. X zero up to XT. Okay, I'm assuming time is continuous. What is the path probability? This is my question. So, I already told you about this, because in the end, here, the stochasticity comes from the noise. So the noise is a sequence of Gaussian random numbers that come one at a time. And this implies that we can write the conditional path probability, like this BXT, given X zero is the initial state. This is the probability of the full path given the initial state. Okay, I hope you're fine. BXT, this is equal to the oxide times a normalization constant times exponential of all this, which is the following. This minus, sorry, I made a mistake here. This you can find it in the book of Schulman on path integral, you can also calculate yourself. I just give you the result here, integral zero to T of this minus this minus this squared. This is coming from the fact that this is Gauss. So this will be gamma X dot S minus F X, that is, minus epsilon FPS to the power two, and then DS. Okay, this is the probability for a trajectory in terms of, okay, remember, this is a way, this is in reality a limit because we are thinking about continuous time is the limit when, when the delta T between every two observations is zero of the X zero up to the XT, and this is the same with noise. So this is the XI zero, the XI. Okay, but these are just the difference of what it's the most important thing here is this fact here. Okay. So, having said this, so this is the probability for a single trajectory. Please, please go. I say that S denotes trajectory. Sorry. No, okay, as is a time instant. So I'm integrating is a dummy variable. I'm integrating as from zero to T. Okay. This is an internal time variable. So I have a trajectory. Okay. Okay. Any other question. The question about the Jacobian. Yes, Jacobian comes out of the race within this too. And it also comes here. Okay, but it is not. It's something that is independent on on the trajectory itself, but if you want to be extremely rigorous. But for what I'm going to do next, the Jacobian is not playing any role. This is the main point. But yet, you just take into account the Jacobian exactly here. It's a normalization factor that depends on the system parameters, for example, on KBT on gamma, and it comes also from the Jacobian. You should put here the determinant of the Jacobian in transforming sideways. Okay. Other questions. Okay. Sorry about that. I can't follow the chat. I don't know why, but I'll follow the chat. Okay. I have the time now. Okay. Yes. Good. So now the question comes on. Given this, this system is larger and dynamics. What is the response? Response function. Response. And I will calculate the response function for the velocity. Okay. First of all, I must say that I'm following very much this PRP paper that I'm saying. And in particular within this PRP paper, there is one section, which is appendix B. So when you read papers, don't forget appendices are very important, especially if there are new results. And you want to look at the mathematical proofs in appendix B, which is called quick derivation of the equality. Okay, so I'm following, I'm trying to follow the shortest path for the quick derivation of the equality. Okay, what is the response? The response function is defined as follows. So what we will do is to compute what is the average velocity in the perturbed system, which is this. So the average velocity will be x, x dot t for value epsilon in the perturbation will be nothing but okay is the average of x dot. So the average of x dot, you can write it as this is the path definition of the path probability for a path. Okay, this is the way we write explicitly the average of x dot. What is the response function? The response function can be found by doing the following by making the functional derivative of x dot t of epsilon with respect to the external force, so the perturbation which is this. Okay, so I will calculate what is the functional derivative of this object, which is x dot, with respect at time t with respect to a perturbation at time s and be very conscious that you can have a perturbation at time s, which is before time t, and this will impact on what happens later at time t. Okay, that's why I'm putting here two times in this. When you do this, you can figure out and it's not so complicated to show that this is equal just you can show it from here, because you can do just the derivative with respect to this. This is an exponential, so when you take a derivative of an exponential, and then you say epsilon f is small, we will then say epsilon is small, you just get from the derivative, just the argument in the exponential. So, all you know what you get when you do this derivative is just this x dot at time t, okay, is this. This is the first product with what comes here without the epsilon, so it's just x dot s minus f xs. Okay, this is just this derivative, this functional derivative. And then we should be careful on what does this average mean, the average is done in the ensemble in which we have epsilon, okay, this is the derivative how much the velocity changes the average value changes when we are setting this value epsilon. So this, that's what we don't like this. Okay, this is the first calculus. So once we know this, we can define the response function as follows the response function at t minus s is excuse me. Yes. So, it's starting with. Yes, I'm using some notes, okay, because here to, in order to have this nice form and that this does not depend on the trajectory. It is more convenient to use the notes. Yeah, and I can, I can integrate this equation and start the notes. Perfect. I can use two different types of integration rule, I am using some notes just because for the sake of calculations. It's going to be much easier because I would just use standard rules of calculus in in migration. Okay, so I'm using some notes. Okay, thanks. Okay, so what is the response function, the response function is just the limit when epsilon goes to zero of this quantity. Okay, how much the velocity is changing when you have a very, very tiny and response. So we will do a limit when epsilon goes to zero of the delta x dot t with respect to an epsilon f p s. This is what I want to calculate when epsilon goes to see, you can figure out that this is very, very evident that way. When epsilon goes to zero, what you can get from here is just the same thing, but here at zero. So we will do this average, when there is no force. So this means that we will have beta over two gamma of x dot t. This is product x dot s minus f x s. Okay, this is in the zero. Okay, so without any right now, I can write this in the following way I will use now. And then the so-called velocity autocorrelation function CV t minus s, which will be the following quantity, it will be the stochastic velocity at time t minus the new velocity times the stochastic velocity at time s minus the new velocity. And this is in a system zero. Okay, for absolutely what I'm just saying what is this V is the average V bar is nothing but the x t at zero. All right. So, when you write this, this becomes x dot t x dot s. This is the first term, which is what it's is appearing actually up here. So, this is appearing here is that x dot s is here. All this doesn't know it. So that's why I'm not putting anything was let me put a circle just to be you want to be very picky. We just put a circle here. Okay, first thing is this, and then there is x dot team with the x dot s with the. And then there is V with V. So x dot average is V. So there will be minus V square minus V square and plus V square. So what you get at the end is minus the average square. And this is in a system. So you see, from this equation, I can continue up here. I'm using this equation. And what I get is that x dot t x dot s is the correlation function of the velocity plus V square. So it becomes beta divided by two. And then the square plus C t minus s. And then we have minus beta divided by two gamma of the rest, which is x dot t. And then we have minus F. Good question. Why is the average velocity dependent off time. Yeah, good question. But you see, when you put here epsilon to zero. What you have is a system that has no, it's a stationary state. What's happening here is that you have a force that depends on X. So it's like a potential doesn't change in time. The particle which is doing like this, but then you are averaging with the distribution of this, this potential. So if I average over many trajectories. This won't depend on time. So I'm doing this stationary average of the velocity. It will depend on whether this is just conservative or not. If you have a particle in a ring, and you do X dot X dot T over many trajectories, you will have a net velocity. If you are only in a potential if this is just partial view of partial X of you, this average velocity will be seen. That's this question. Okay, I mean, I still am not clear, but I think I can try it. Okay, okay. Okay. Anyway, okay, so this is the average of a, over many, many, many infinite trajectories in a time T of a system that has no time dependent. So it's not. There is not here as an independent driving is either an equilibrium system or a non-equivalent steady state. This is very important that this process for epsilon equal to zero. I just remarked this for epsilon equal to zero. The system is either an equilibrium system, equilibrium dynamics or non-equivalent steady state. Okay, this will depend on the nature of it. If F is just the derivative of potential, so F is just minus partial X of you, we will have an equilibrium dynamics. If F is the derivative of a potential plus an external force, we will have an, the amplitude system is a non-equivalent steady state. Okay. This is for epsilon equal to zero for epsilon positive, we have a given system so we have either an equilibrium system that is driven or a non-equivalent steady state with driving. Either equilibrium or non-equivalent steady states, steady state plus drive. Here, X dot of T depends on time. So this partial T is non-zero. Here, because we are in equilibrium or non-equivalent steady state, the net velocity equal to zero. Okay. So this satisfies a little bit the question. Let me advance because otherwise I will not be able to explain the key result. So, okay, so we continue with this. Just be aware that here there was a gamma that was killed in this part. Okay, in the second part it's surviving. Very nice. So this is the response function at T minus S. And now what I will do is I will sum the response function at T minus S plus the response function at S minus T. Okay, what is the response at time T if I apply a pulse at S and plus the opposite. So if you take this equation and sum T minus S and the case of S minus T, you get first of all the following beta average velocity square plus C T minus S. Okay, here I'm using the fact that the velocity velocity autocorrelation function is even an entire reversal. So C of T equals C minus T. So C of T minus S equals C of S minus T. That's why there are two factors here. Right. This is one. And then the second part has TS and ST. So we have minus beta over two gamma. X dot T, ST, okay. STATONOID F says plus X dot S STATONOID F XT. Okay, so this is what we get and both are in the unperturbed system. This is the average of X dot times F. It's not X dot. So this is something else very important. Okay, so now what I'll do is I will do the limit when S goes to T from below. Okay, if I do this, I get something like two are zero. And because S going to T minus or S going to T plus S going to T basically makes that T is equal to S. So we are evaluating the response at time zero becomes beta B square plus C zero the autocorrelation function of the velocity. These are correlation factors of velocity, very important. Minus and then T equals to S on the right is just beta over gamma X dot T STATONOID F XT. Okay, at the same time. And then this is at C. Okay, so I don't need this part. And this is the main question. So in reality I could stop here and tell you this is a fantastic equation which relates a dissipation with breaking off the partitioning situation. But if you see this, this equation, you won't see it very intuitive. So just keep this in mind. And let me just introduce something quite relevant that is the Fourier transformers. So, and I think there was a there was a question about it. So let me try to explain now. Okay, so this is our main equation and I'll try. I don't know if I can copy this. I will make a try if I can copy this equation. I manage. It's always quite complicated to learn a new technique every day. Edit. Now we cannot make it. This is all so fun. Okay, so I take this and I copy. No, I cannot copy. Okay, so I go by hand. The result was the following. The result was to R0. The response factor of the velocity time zero and equals beta. You have square plus C0. Not correct. C0 minus beta over gamma of X dot t. Okay, in the system. This is in the system. This is in the system. This is the equilibrium correlations correlations in the system. This is the response when you weekly perturbed the system, and this is an average in them perturbed system. All right, so now I introduce the Fourier transforms in this in the same way. In the paper, Fourier transforms. So I'll first define, for example, the Fourier transform for the autocorrelation function, which is a, I'll define it as follows in this convention DT of 2 pi minus infinity to plus infinity. DT of Ct. E i omega t. This is in the way they define Fourier transforms in this. Okay, just be aware of one thing. That is that, and this is in principle complex, because this is real, and this is a complex number, but C of t is symmetric. So it is even with with respect to the reversal and this is cost omega t. Plus I sign omega t, which is all. So if you multiply, sorry, even by even. Yes, sorry. Okay, someone is calling. If you multiply even by even, this should be no zero and if you multiply even by all the integrals will be zero. So in the end, here we will just get something is DT of 2 pi. C of t. Okay, this means that city le omega is real. That's only a real path. Okay. This is the first thing. The second thing I want to say here is about the response function. So the response function the freedom form. I'll call it like this, the same way BT over 2 pi out of T. I omega t. So it is not to guarantee that the response function is real in general it has both real and imaginary parts. This is very important for you to realize. And when we were doing this limit here. When we are doing this limit, we were doing as going to T minus in reality. When we hear when we do the limit as going to T, we will have in reality, this is not very well explained, but this should be towards zero realities are zero plus plus or minus. If I am going to be very, very mathematical with this. Okay, so when you do this, and you have this. So, of course, now the R t minus s will be the inverse freedom form. So it will be the omega over 2 pi are put it on from omega e minus omega at t minus s. Okay, this minus infinity to plus infinity. So what we are interested in is are at zero plus plus are at zero minus, and this becomes, you can show very simply, this becomes the integral the omega to pi. Please realize that here t minus s t minus s, if we put the minus is equal to zero is the integral of the full is this integral with e to the zero which is one. So the R zero plus plus or minus becomes this times two, and this is the real part. Okay, when I put a prime is the real part of the Fourier transform of the response funds. Right, this is one thing. And another thing is from here you realize that C tilde, sorry, not C tilde, C of zero. The result will be this one, but now I'm going to write it in a more fancy way, because I'm going to use this free transform and see zero is nothing but the integral the omega to pi of C tilde omega. Okay, so, of course, this is, if I put here, omega, sorry, t to zero. It's not here but in the inverse free it's the same thing as here, you get this one. So, this means, this means that this part. This part here. All this part I will change it by this. And this part. By this. And I'm just missing one part for the quality which is this, this will be a X dot F. And it is not so difficult to show that X dot F. Nothing, but mind the heat dissipate in the system. Okay, in the amperter system. Of course the amperter system. The system is either equilibrium or non equilibrium state state if it's equilibrium. Here q dot is zero. There is no dissipation of heat. If it's only clear in the state it is not q dot is negative. Okay, so this is equal to this. It is not so, I mean it's not just one and I need to prove this because what I showed in my previous lecture was that this is equal to minus F X dot. Okay, but you can show that this is equal to X dot F. And I leave it to you as an exercise. This is a nice exercise. So, you see that in this equation we have R zero, which is response C is correlation and this is dissipation. So, when you put all of them together, you get this very beautiful equation, which is the form the heat dissipated in the amperter system is a pre factor which is the friction coefficient times v squared plus the integral from minus infinity to plus infinity of the omega by two pi of the correlation function of the velocity. So, we have omega minus two KBT, the real part of the velocity of the freedom of the response function of the velocity and in the aspect. Okay, so look at this equation it is really beautiful and really insightful, because it says the following. The equation of the amperter system is related to two things. First, the net velocity, this is like the drift, like gamma v square is when you have a particle in a fluid moving at a constant velocity, its dissipation will be gamma v square. This is the naive path. And this is a measure of the extent of the violation of the fluctuation dissipation relation. So here is an equality connecting dissipation, because this at the end in the system is the Android production rate in units of KB connecting with violation of the fluctuation dissipation relation. So why am I saying this? Well, because there is, and when we have q dot equal to zero, which will be equilibrium systems, this implies, okay, also the V bar is zero. And we have see the omega equals to KBT r prime omega. And this is the so called fluctuation dissipation relation. And it was proved by Kubo. Quite some years ago, you can find it. Okay. Another Japanese Japanese are very strong in this topic report. This in the year 1966 is a very important and classic result in statistical physics. Okay, so if you go to the paper of Kubo, you won't find it written like this, you will see here the imaginary part, but this is because he writes, and very many people write often the correlation function for for X instead of for B. So that's what it brings me to the conventional conventional formulation of the formulation of of the fluctuation dissipation relation. In fact, so what we are doing now is auto correlation function of velocity can be written as this for the transform of V, for the transform of V at minus omega. This you can show that this related also today. So we have a return from effects in this way, minus i omega, return from minus at minus on. And this shows that this is imaginary unit i times minus i is one, and then omega squared, so it's omega squared x to the omega x to the minus omega. So this means that the auto correlation function of the velocity at omega is omega squared, the auto correlation function of x at home. Okay, this is a very simple result to prove another analogous result you can show is that the response function in the velocity is. This is what what just is behind all this is just i omega, the response function in x at omega. So this implies that the imaginary part of the velocity to creation function is omega times the real part. So this is the actual function in x. Okay. Yeah, I think I made a mistake here. I think, okay, this is the real part. And here, okay, here we are relating imaginary real, and there's the same analogous with real and imaginary. So all in all, if you want to write this one. You can find out what happens in kubo's paper kubo doesn't write it like this, who writes it for x and for x, the conventional formulation is like this, c x omega equals, instead of two kbt will be an omega squared to kbt divided by omega squared imaginary part of the response function of x. Okay, so you can also write her as as a, instead of in terms of response function of x of velocity with response function of x, the only difference will be that here, you will have to put x, and here, omega squared. This is the response function of x and imaginary part. Okay, so if you go to a literature, you will find both formulations and they are q. All right, so with this I am done with the result and I don't know if I have time to the other concept I wanted to explain today, which is also related to the question response. Okay, I'll try to go through and if I don't manage and I explain you more later in the second part of the lecture. Okay, this is a very also exciting topic, which has a very funny name, which is the following, is the Frenzy. The Frenzy is a concept introduced by different authors and, for instance, Christian Meijs has now a review of this topic in the physics reports. 2020. And there are also very good papers about this also called, okay, this is now the most accepted name but there are other names such as traffic. There's also activity, etc. Okay, and there are other relevant authors in this, in this topic, which, for example, Marco Bayesi. He has very, very beautiful papers in this topic. Why not as well. Was the PhD student of Mr. Meijs and Urna Basso as well as very nice papers. I did one paper with Pierre Paolo Vivo as well. So there is growing community working on this part. And, okay, let me try to explain what it is. So, I told you before that a priority for a trajectory has a form, and typically we can write the priority of any trajectory as follows. The priority will be so-called action will be a functional of the trajectory and P0 will be the probability of the same trajectory in another process, it will be a reference process. Okay. So, in other words, I will define AXT as the algorithm for the probability in a reference process to see a trajectory divided by the probability in the process of my interest to see this trajectory. All right. So, one particular case is when this one, when this one is the probability in a conjugated with the time reversal. If we do this, A will be the entry production, but this is more general what I'm going to explain. So, first of all, I can show you that one can, if you want to do an average of XT, X of time T, this will be nothing but, well, in the measure P, this will be, you can do it like this, average of all trajectories of X of time T probability of the trajectory. Here it could be something else, it could be a function, it could be anything else. And you can also write this as the average of XT with E to the minus A. Like the way you change measure with P, C, okay, XT. Then you can do the average of anything in one process. And using this A, you can do the average in the other process. In other words, in other words, A is just a change of measure. Now it comes, what is the so-called frenzy? So what I'll say is that I will split A into parts, one that is even under time reversal, and the other which is odd under time reversal. So, I will say that A, it will be D minus S divided by Q. Okay, S is the same thing I explained you before, which is the entropy production, and D is this new object that is called frenzy. But actually, instead of calling D, because I use D for diffusion coefficient, let me use another word, another symbol that I like more for this object, which is phi. Okay, phi is the frenzy, and S is the entropy. Okay, so what are the two things that characterize this? So the entropy, as we said, is a functional that is odd under time reversal. So you measure the entropy production of the time reversal trajectory, you get minus the entropy reaction of the original trajectory. However, so this is the entropy reaction. And what is the frenzy? Frenzy is the symmetric part of the action. Okay, I did everything with D, and now I have to trace back. The frenzy is as follows. When it applies to the trajectory, it takes the same value as when it's applied to the time reversal. This is called the frenzy. Okay, what is called frenzy? Well, I can really, the best is to look at examples, but it is because it's related to the number of jumps that are happening in a trajectory. So the entropy is related to the reversibility, but frenzy is related to how many jumps in any direction are happening in your trajectory. Okay, but this is easier to see with examples. Okay, so now, knowing this, what we can do is compute the probability for a trajectory divided by probability for a time reversal trajectory in a non-mechanical state. Okay, this is very easy because now we say this is e minus phi xd minus s xd divided by kb. And now divided by e of minus phi of the time reversal trajectory minus s of the time reversal trajectory divided by kb. First of all, phi of xd and phi of xd reverse is the same. So this goes away and you just get an e today. Okay, I think I was missing a factor two here. I'm missing a factor two. I think I mistake, I am mistaking the sign here. I'm mistaking the sign here with a plus. Okay, I think with a plus this becomes s xd divided by kb. So, and this brings back to the definition that I introduced in my course. And this is s top, okay, s xd is kb times the logarithm of the probability for a trajectory divided by the probability for time reversal. Okay, this is consistent with this. All right. However, what if now you make the product that you do probability for xd time. Yes. Are there results in the conventional, the same as Kramers-Kronig relations. I must say, I don't remember. I don't remember which were the Kramers-Kronig relations.  Who is being disposed to you? Okay. Good to know, but I don't, I didn't remember this. Good point. Thank you. All right, this time. So now if you make, instead of the ratio, you make the, if you multiply this time this, the under reflection cancels because this is equal to minus s xd. And the only thing that survives is the frenzy. So you just get minus 25 xd plus. Okay, so in other words, the frenzy, it's for a trajectory is related to something more weird, which is minus the logarithm of the square root of p xd times p xd bar. Of course, you see, this doesn't make any physical sense because you see that the distributions, path probabilities have units. So the only thing that makes sense is the change in frenzy. So if you do this, the change of this quantity, you do d. But as many quantities in, in, in thermodynamics, if you do d at t minus d at zero, what you do is basically you will have minus the logarithm of the square root of p xd, p xd bar, and then p x0, p x0 bar. Okay, so things that make sense are, make sense are changes in, in frenzy. Okay, so this is not so important as what I'm going to say next, which is that the fact that, okay, as I said before, the average of any functional of a trajectory, p xd, in the, any, with, with, with measure p, so we have a physical process, we want to know the average of the work, we can write it as the average of e to the minus a xd. So this is the action times the same functional in a reference process, whatever process we want. Okay, and this a is defined between p and pc. This implies that you can show this easily that if you have a weak perturbation, you will have the following that this will be a similar or equal to. So we'll develop this. Sorry, there's some noise in the background. We'll develop this for a small. So then this will become one minus a times omega zero. So this will be and omega at zero minus eight. So close to for x and small, this will be the first contribution to the response. And this you can show if you do, for instance, the response is in this way, the epsilon of omega xd at epsilon. And then this will be the limit when epsilon goes to zero of omega, evaluated a trajectory in epsilon minus omega xd in the amperter system, divided by action. This is the definition of the derivative of how much it changes an observable when we have a weak perturbation when absent. So from this equation in the middle, this is similar or equal than minus the functional. Times a xd. This is trajectories. In. So this means that we can write the response. We're studying this way as one half. Omega times s s prime epsilon in epsilon minus omega times five prime epsilon. So it means that there are two contributions. This is the derivative with respect to epsilon. There are two contributions. One is an entropic entropic term. And the second is a frenetic term. It's a term that is proportional to the frenzy. And this is very important. It was a very important insight of this theory that entropy and frenzy for the response of any observable are equally important. One is one half and the other is, it's a factor one. So for instance, if you take this example, omega goes to one. Okay, this is the most silly. Edgar. So I think that is a question so. Regarding the absence of the a should be proportional to absence of course you should have. Epsilon time a no in this formulas and not necessarily not necessarily because. Okay. Here you should have the A. The epsilon. Yes, yes, yes. Okay, that's the point that's the point. Sorry, I forgot that you're right. You're right. In the same way we were doing the derivatives function derivatives as here. Okay. Yeah. Okay, that's, yeah, that was a typo. Thank you. Very, very good point. Okay, that's why here that is s prime of some these are functional and very active perspectives. Omega equals to one, this is one valid functional is the most silly you can think of this implies so of course the derivative of one is zero. This implies this equation. Okay, this is the central equation from this theory of, of frenzy. This is the key result. So you put seat one, the derivative one is zero, and this one, but what you get out of it is the following. You get that the entropy minus the frenzy. You can show this in a quality very simply. Therefore, you have something like this in the systems get very equal than five. So it is like a tight bound to the entire production tighter than the second one. It's not only that the entire production is getting zero, but the entire production is also greater or equal than the frenzy divided. This is sort of a second. Okay, but let me go further because sorry Edgar. Yes. It will be twice the frenzy. Yes. Yes. I made a mistake here. Yes. Yes. Yes. Exactly. This is twice. Correct. This should be twice in the way I defined. All right. And let me just wrap up with this. This is probably one of this and thought as the most and the best application of the results, which is when you use as omega, you use a time integrated current, which will be called gay. This will be a time integrated, integrated current. It could be, for example, the position of a colloid could be a work up to time t source of time integrated current because you are summing the work over a different time intervals. Heat into reproduction, etc. It could be also your mark of process with four states and you count, for example, every time you jump from one to two, you count plus one. And when you jump this way, you count minus one minus one. This is also time integrated current. So you will have jumps when this is happening. And minus one when this is happening. And at the very end, if there are drifts, you will have this will be a time integrated car. You can apply the same quantity and the same relation sorry to the response of a current, and it will be written. So in the same way. So you will get this same equation, but now instead of omega, you will have any current for Markov Markovian system. The response of a current will have this term that goes with the the entropy and this term that depends on the furnace. If the current is the entropy you will have here a sort of an S square. So if, if J equals to S, this is a positive contribution, but this can be negative. And this is very important when you consider examples and one example that is very much. If you have this J, I'm going to simplify notation very much. You will have the J with respect to epsilon is J as prime divided by two minus G. Okay, so why this term is important. This is being discussed in the last years. Very nice papers one I can highly recommend you with examples. This is a version of one of us and mice, one of us who increased and mice. This is not in a very well known journal but it is very, very well written, we like it is J. I think it's from an Indian conference. J fees, come cities. In 2015, some very, very, very nice paper, I really like the examples, it's good for you to read it, because you will see the fancy can be easily measured in some modes. And, and then discuss for instance, the following system which is an active matter system. For instance, you have a system like this where there are some obstacles, and you have some swimmers that are trying to pass these obstacles and they have a drift. These swimmers are of course in an active, sorry, in a thermal bath, but the swimmers are managing self proper particles. This is paradigmatic model of active matter. This type of dynamics can be described with simple stochastic processes actually. And what happens here is you would say, okay, what happens when I increase, let's say the field. So imagine you put, for instance, a concentration gradient of food. And here it would be actually the opposite. So there will be concentration of food along x like this. So bacteria would like to go where there is more food. Then, and this will define a parameter which will be the slope of this, of this concentration gradient, and this will determine the net velocity of this, but you know, at least, if there is no this objects, these obstacles. I think that when I put more and more gradient of food, the bacteria will move faster, but in some experiments has been shown that, okay, first the net current of bacteria. When you increase epsilon increases. This is actually the effect of this part. You can, you can see, but then at some point when you. This will be another situation with higher value of epsilon or concentration gradient. It can happen that there are there are crowding events. So the bacteria are not helping each other and they, they are and they are obstacles to themselves. So it can happen that there are trapping effects here. Okay. The bacteria are passing. But when there are many, many people trying to go, there is here a trap. So they are trapping effects. These trapping effects can be captured by the simple models and you can really go to this paper and check an example. And this is a manifestation of of frenzy because here, there are a lot of jumps in all directions, which can be quantified with a with a quantity of this. And this generates that when action is large, you can have this type of response. So this is called. This region is called the region of negative. This is called the difference and response response, and it can be understood the best way by having this phonetic term in the, in the description without this. It is not possible. So this type of dynamics will have a monotonous increase. This would be without tendency. This will increase the current like this, but never show distance effect. So this is very, very important for active matter system system to the extended volume, etc. Okay. Again, let me just remark that here there was a field, so it's like a field pushing in the side. Okay. I don't know how much time they have, because I wanted to show you experimental results. Some slides. Maybe we can take a break of five minutes and then a good point now. Yeah, it's a good point. It's a good point. Let me know when do we stop the recording just to be sure. Okay. I stopped the recording now.


Raymond Burr as Captain Lee Quince, specially transcribed tales of the dark and tragic ground of the wild frontier, the saga of fighting men who rode the rim of empire, and the dramatic story of Lee Quince, Captain of Cavalry. Mr. Saibotts, Mr. Saibotts! Got fire, sir? Seize fire. But they'll think we have quit, sir. I hope they do, Mr. Saibotts. Yes, sir. Captain, what about your leg? It's fine. Now pass the word, Mr. Saibotts. Right away, sir. Seize fire. Sergeant Gorse! Right here, Captain. Give me a hand. Behind that boulder. Yes, sir. Can you bear weight on that leg? I can get behind that boulder. Are you easy now, sir? Don't know when I've been more comfortable. If you're dry, I got part of a canteen here. No, no. Save it. You'll need it. Captain, there's a clearing about 100 yards back. We got the horses back there. Harrison and me together. I'll get you there. I'm fine here, Sergeant. You'd rest a might easier there, sir. Maybe you ought to take up nursing, Gorse. You're developing a fine manner for it. Well, sir, I reckon it's all this easy living. It's kind of got me soft. Go back to your position, Sergeant, and dig in. Flank in the canyon. This way, you must be figuring them engines will try another run-through. No use. 40 men spreading thin against 200, 300 to wrap a hoe. Yes, sir. Go on back, Gorse. Send Lieutenant Cybert's here. Right, sir. Lieutenant? Lieutenant Cybert. You set for me, sir? Men have their orders. All down the line. How about ammunition? We've enough, sir. What's enough? Well, sir, I think. Enough to lead a charge down the canyon and run them off? No, I don't believe so.  That would depend on the size of the assault. Mr. Cybert's there at least 200 to wrap a hoe fire in what's left of our camp down there. I want to know if we can hold these positions. Not long, sir, not if they turn on us full force. Then we don't have enough ammunition, do we, Mr. Cybert's? No, sir, I guess not. What do we do, Captain? Sit here. Hope they think they wiped us out in their last run-through. That why you ordered the men to hold their fire? That's why. They're riding off, Captain. Don't look like they counted us too important, Captain Quint. That complaint, Sergeant? No, sir. I was born plain insignificant. I mean to stay that way. Help me get the Captain back to the clearing, Sergeant. His leg needs attention. Wait a minute, Mr. Cybert's. We can't waste any more daylight. You have six hours of good riding left before nightfall. Sir, you can't ride hard. You've lost a lot of blood. The patrol's going back to the fort under your command, Mr. Cybert's. Harrison and I will take it in easy stages. We can't leave you here, sir. Mr. Cybert's, you'll reach Fort Laramie tomorrow night. If you're lucky. Major Daggett needs a full report as soon as possible. Captain Quint, sir, we could strap you in your saddle. No need for that, Sergeant. I'd slow you up, and Major Daggett needs that report. Now, send Harrison to me. Prepare to move out. I know this country better than Harrison, Captain. Mr. Cybert's will need you, Gorse. This will be good experience for my orderly. Yes, Captain. Patrol, report to the clearing. Check your mouth. Prepare to move out. Good luck, Mr. Cybert's. Thank you, Captain. Report to the clearing now. Good luck to you, sir.  There's fresh water ahead to the left there, that clump of trees. You can't go on just water, Captain. Well, it'll help. You can water the horses, too. Not much sunlight left. I hoped we'd find a settler where we could bed down for the night. This is a wrap-a-hold country, Harrison. They routed out most of the settlers. They sure got away acting like they owned the place, don't they, running off, whiting in. They were here first. I guess they got a funny idea that makes it their land. If we'd have had the whole company back there, we'd have run through them, sure. Show them whose land it is. Well, it would take an awful good company. Those were dog soldiers leading that raid. Dog soldiers? Toughest fighters in the tribes. Handpicked for the Darren, Hote Mintonio. How's Epson? That's the Cheyenne name for dog soldiers. But most Plains Indians have a select ban like them. Sue and the wrap-a-hold. Oh. Guess we were smart to take cover in that canyon, Captain. Yeah, I guess we were. In through the trees, Harrison. Good grass there by that stream. I'll be right there to lend a hand, sir. Stand steady me as I come over on this leg, Harrison. We'll bear weight. Yes, sir. All right. Oh. I guess I'm weaker than I thought. Don't you worry, Captain. I've got bull strength. Here, let me help you.  There. How's that, sir? All right, Harrison. Thanks. I'll get you some fresh water. You go on all the way. No, I'm all right. I'll ease those cinches and leave the mounts down to the stream and water them. Go on. I just need to catch my breath. Rest a bit. I sure wish I could make you right comfortable, sir. In bed with food and all. I know. Mind the horses. Yes, sir. Harrison. You awake, sir? You shouldn't have gone to sleep. You just didn't have no choice, Captain. Once you drank that cool water, you was plumbed through. What's that? Horses? No, they're not making a sound, sir. I hobbled them before I sat out. Shh. Wait a minute. His horse is all right. Sounds like they're riding this way. I sure don't hear a thing. Could be you got some fever, Captain Quince. Some folks hear things when they got fever. You keep listening and you'll hear them. Yes, sir. What I was going to say was, while you was resting so easy, like, I sat out and took a look around. There's a cabin, sir. Not more than a couple of hundred yards away. They're coming from the north. Yeah. Yeah, I can hear them now. Could be cavalry, Captain. They move south long ahead of us. Look, look through the trees. Engines. Dog soldiers. Captain, we better make for that cabin. No, no, no. We're better off here. The trees will hide us. Unless they already know we're here. Lifeline Harrison. They start through the trees. Fire. Can you handle that carbine, sir? Yeah, I can. They're making for us, sir. Sure, we're seeing them. Wait them out. They're right at the tree, Captain. Easy, easy. I'm sure not keen on them seeing me first.  On the tree line. Going on south. I can't make them out anymore. You see them? They've gone on past the thicket. I can't see them. That was close, sir. Still close. They stopped. You want us to get our horses, sir? I do not. They ain't moving as I can tell. I counted six, Captain. Six? Where is it? Near as I can tell. That's where they are, sir. It's that direction, not more than a couple hundred yards away. I tell you, sir, that's a welcome sound. You were talking about a deserted cabin, Harrison. Not a living thing around. Likely been that way for years. You'll have to near carry me. I can't mount again. Just you do worry yourself, Captain. I'll take right good care of you. Not ten more steps. You can make it, sir. I don't know, Harrison. You just lean heavy now. We'll get there. You make out fine. You don't come no closer. Ma'am, there's no cause for shooting. We're your kind, white folks. I'll judge about the shooting. My day is him.  Ma'am, I'm not even asking for a hand, but I've got to put him down inside. Oh, me? The 2nd Cavalry, Fort Laramie. Harrison, I think I... Captain Wimps. Captain, you all right, sir? Is he dead? No, ma'am, not yet. Would you just try to help? Put that shotgun down and help a body? How are you on food? Oh, a tin of jerky, maybe. Blankets? Shot? You got anything?  That's what I got. And I'm lifting him up right now and taking him in there. How much money you've got? I don't know. A little. Hand it over. Once we're inside.   I don't rightly know how you stand yourself. His money, too. I'm bringing him inside now. Not for long. For this little bit of money. You're another weekend, are you? My arm's full of logs. Put him by the stove and quit your whimpering. Soft belly's a lot of you. How is he? Ask him. He's only pretending to sleep. Captain? You can't fool her, Harrison. Don't try. You gave me a fright, sir. I wasn't sure you was going to come around. I'll be all right. Did she get the bullet? Yeah. Yeah, she got it. I see new cubs are tougher hide. Captain, she took everything we had. Jerky, guns, money. What kind of white woman is she? Best quiet, him. I buried four husbands, 20 children. Couple of strangers don't faze me. Better get some sleep, Harrison. She says you watched me all night. You sure you all right? I'm fine. I go to sleep. That's an order. Yes, sir. Outlived them. But they were men, regular men. Didn't go around siring every other man. And ma'am and women folk. Yeah, they weren't in the army. Indeed they wasn't. No call for the army in those days. Mr. Palfrey and me lived as peaceable as could be with the Indian folk. We come here peaceful, me and Mr. Palfrey. When was that? In the thirties. Things were different then. Mr. Griffith, Mr. Albright. Even later on with Mr. Netherton. Yeah, there was lessons to learn in the west in them days. Eat the porridge. Much obliged, ma'am. Hattie. How's that? Hattie, I say. Hattie Palfrey, I go by that. I can't abide being called ma'am. Oh. I lean toward Hattie. Hattie, you live alone. Middle of Arapaho country. I didn't manage. Come by your guns.  Sometimes it's food, sometimes robes. I'll take what I can get. It don't matter to me if it's settlers or scouts or army folks. Or Arapaho. Folks don't find Hattie unless and they lost. Or half dead. Open your eyes to see what they part with. You always greet them with a shotgun. If need be. Before we came here last evening, a band of Arapaho braves stopped here. I got good errors. Strong bows. Rifles. Even a couple of Buffalo guns off a hunter broke his leg down the canyon. They got a camp near here, the Arapaho. You got something more to give me? Harrison said you cleaned us out. You're getting full measure. What's going on? My ring. Hey, you can't have that. But I got it. I ain't had a pretty first spout and that's a fact. A man ain't safe sleeping here. I bought that ring myself in St. Louis. It'll fit fine once a wrap some twine around it. Bought his fingers thin with age. Next you'll be picking my bones, you old buzzard. Mind your tongue. Give him his ring back, Hattie. Not likely. Captain Quince, what says you can't hit old woman when she's so contrary? Oh, for one thing, she's got a shotgun. I'd soon use it as jaw with you.  I do admire a pretty. Clean forgot the bright feeling they give you. Our guns for the ring, Hattie. That's fair. I come by your horses down the creek this morning. They're grazing on my grass, swillin' my water. The prettiest to pay for that. Captain, I better seat in them mounts.  I'll seat the horses. You sayin' he can't leave the cabin? Not alive, he can't. Why, you... She means it, Harrison. I do that. Now, I've got tendon to do. Outside. No need peering around for your rifles whilst I'm gone. You won't find them. Captain Quince, we gotta move out of here. No, we need horses to do it. Here, you steady me. You ain't fit to stand yet, son. Just you stand still. Yeah. We'll have to crawl once we're outside anyway. I can manage to the door. The lack is not she just crouched out there waitin' for us. It's worth a chance. We may not get another. Yes, sir. Get the door. Get on your belly. Move small through this brush toward the stream. Can you see her? No, sir. Stream, how far is it? A good 50 yards. Maybe more. Do you see the horses? Not yet, sir. Flatten, Harrison. It's a wrap-hole. She'd come out of those trees by the water, sir. She's talking to them Indians. Don't even breathe. She's going back toward the trees. Yeah. But they ain't a sign of our horses, sir. No sense trying to go it without them. We'll have to go back. To the cabin? Without guns or horses, we're no good out here, Harrison. Come on. What do you reckon? She's done with them, Captain. That woman? She might have eaten them. I'll steady you for the walk, sir. I'll give her a lot for some of her strength right now. You think I could find them alone?  It's no good, Harrison. We'll have to wait. It's about as foolish as it can be, ain't you? Crawling around in the brush out there. What's I.D.? Got no guns, no food. There's a wrap-a-ho all around. You don't think too good of your hides, do you? Where are the horses, Hattie? The color's coming back some. You heard the captain. What about the horses? They come right high in these parts, young'n. I could get me a parcel of things, trading horses. You're real friendly with the wrap-a-ho, Hattie? They treat me good. Of course they know Mr. Periphery and me come peace-able to their country. Not to run them off what rightly is their land. You never seen them at the killing. White women, babies. No matter to them. You ever ask yourself who started it all? I've seen it happen. The whites and their guns moving in. It wasn't pretty work they did. Women, children, too. Hey, they let you live here. There's got to be a reason.  Mr. Periphery and me, we took him in, attended him. He was a young chief then. Standing bare, never forgot. Not in all these years. He's as near to a relation I got. You're from another age, Hattie. You and standing bare. You got anyone? A wife? No, no wife. Mother then? I got her memory. Periphery rings, that's all. Not on you. I'd have found him. They're back at Fort Laramie. Whoa, that's a pity. Yeah, that's a pity. Think you can get the guns without waking her? I can't make out good with no light. But I can fight an old woman in the dark. Not long till, son, this may be our last chance. Easy now. Now that's just close enough. Stealing up on an old woman. What kind of men are you? I don't mind shooting a bit.  That's what you are. Seeing in the dark, hearing. You're no kind of woman at all. Maybe not now, young'em. But I was once more woman than you could imagine.  Someone's coming. Come earlier than I said. It's probably them same engines again, Captain. Now get back where you belong. Over there, to your beds. Captain, I had my gun. I had my gun. I'd blow us all up before I'd let them take us. I know. Maybe I'd help you. She's out there right now. Trading for us. You know that. Get ready, someone's coming. That sight of beef will last me the winter and more. Gift for my friend, standing bear. For us, Hattie? Now then, you're worth a sight more than that. They're riding off, Captain. You sparing us, Hattie? For what? Sparing you. You didn't tell them about us? They didn't ask. You acting human so sudden, like? After they're scaring us to death.  That's a fact. I'd soon keep company with a ball in calf. I'll tell you I'm bone brittle with you two around. Well, we'd be glad to get out of your way. Trade for guns, our horses, and a cut of your new beef. I've been thinking about them earbuds. I've been thinking about our guns. Under my bed. These two horses are saddled at the spring. Now, them earbuds. They're back at Fort Laramie, Hattie. You mean to ride back with us? I gotta watch who I'm seeing with. Yeah. Well, there I might get back your way someday. You do well. I'll bring them to you. For these old ears. Here, young'em. Pretties ain't for me, not no more. I'll take my ring, ma'am. And thank you. But I ain't even pretending to understand you. If you was regular men, you'd find yourself a woman. Pretties don't belong on a man's hand. Locked up at no fort. Now, as women folk, it gives us that bright feeling. Having a pretty. Next week, another transcribed story of the Northwest Frontier. And the troopers who fought under Lee Quintz, captain of Cavalry. Anyone who has ever been helped by the Red Cross can tell you how important help is when the need arises. And everyone who is a member of the American Red Cross can tell you how much deep satisfaction comes. With knowing you'll help provide aid in an emergency. Make your investment in humanitarianism soon. CBS Radio urges you to join and serve your local Red Cross chapter. Your membership and your contributions make all Americans neighbors in a time of need.


Good day to all coronavirus survivors, those of you who are watching this video, it's a third in a series about borderline personality disorder, the inner dynamics and interactions with narcissistic partners, narcissistic intimate partners. In the first video titled Borderline Woman, a dissociative secondary psychopath, we reviewed the current literature, current academic literature and the most recent thinking about borderline personality disorder. This provoked a hailstorm of negative reactions from various self-appointed activists, self-styled experts and borderline women of course. We are not psychopaths, you are wrong, we don't dissociate etc etc etc and I'm afraid this flies in the face of all the evidence we have. Currently the thinking is that borderline personality disorder is a manifestation of psychopathy in women, it's a female manifestation of secondary psychopathy. Mind you, not primary psychopathy, not the type that is described by Robert Hare among males, but another type of psychopathy, facto 2 psychopathy. Also, the current thinking is that borderline personality disorder involves dissociation to such an extent that it can be safely and reliably described as a form of dissociative identity disorder or at the very least as a form of extreme dissociation. It seems that modulability, lack of impulse control, externalised aggression and other manifestations of borderline behaviour have to do with alterations in self-states. Borderline has a diffuse identity, it has an unregulated or dysregulated core of emotions and so it's not quite sure about its own boundaries. It uses its intimate partner and other people, but mostly its intimate partner, to regulate her internal environment. The borderline's dependence on her intimate partner is similar to the co-dependence. And her terrified abandonment anxiety, the reason she is absolutely horrified by the mere thought of being rejected and abandoned, is because the intimate partner fulfills very critical ego functions. The intimate partner is the borderline's ego in effect. And this is very similar to the narcissist. Indeed, as any therapist will tell you, as any psychologist will tell you, and as many borderline women will tell you, borderline personality disorder involves a very pronounced dimension of grandiosity and narcissism. So the two disorders are very much alike in some senses. For example, the borderline also has the equivalent of a false self. In many respects, both narcissistic personality disorder, borderline personality disorder, may be forms of multiple personality disorder, as it used to be called, or dissociative identity disorder, as the current label is. The second video that I've made had to do with the borderline's reaction to abandonment and rejection, especially by a narcissistic partner. The borderline tells the narcissist, I know you're going to abandon me, I know you're going to reject me, I know you're going to abuse me, and I'm going to do this to you first. I'm going to abandon you preemptively. I'm going to reject you before you reject me. I'm going to dump you before this happens. So this is the second video. And today in the third video, I would like to describe in great detail the sequence of external events and internal processes inside the borderline and outside, as she encounters or expects rejection and abuse. So stay tuned, it's going to be stormy. It all starts when the borderline thinks that she is about to be rejected or abandoned. Now pay attention. A borderline reacts to three distinct states. She reacts to actually being abandoned and rejected when the evidence is indisputable and discernible. She reacts to behaviors of the intimate partner, which she misinterprets or interprets as rejection and abandonment, even if these behaviors are utterly innocuous and innocent. For example, the partner has to travel in business, or the partner goes out with friends or the partner is not available when she wants him to be. So this is interpreted as rejection and abandonment or even abuse. And she also reacts to anticipated abandonment and rejection, even when there's no hint or sign of it, even within the love-bombing phase. There's always this thought circulating in her mind. It's going to dump me. It's going to abandon me. It's going to reject me. It's going to abuse me. It's going to hurt me. It's not safe. It's going to hurt me. In the past, her personal history is replete with dozens of such cases. She is simply extrapolating from her personal experience in the past into the future. So she anticipates abuse. When her anticipation leads her to aggress against the intimate partner, for example, to abuse the intimate partner, of course, this creates a vicious cycle and the partner withdraws. The intimate partner becomes less intimate. We draw, isolates himself, finds alternatives, becomes an alcoholic. I don't know what develops addictions. And this is again perceived as rejection and abandonment. The borderline brings on, brings about her worst fears. Her anticipatory and preemptive behaviors are the ones that bring about what she had been dreading all along. And so if the partner is a narcissist or a psychopath, usually this will involve abuse because narcissists and psychopaths or especially psychopaths have something called reactance. They, they can't regulate their impulses and they are defined and they react with extreme abuse, disproportional abuse in a way they catastrophize the abuse to the slightest frustration. So they have a very low threshold of frustration and very high aggressive response. So when the borderline does this, when she constantly pricks, she constantly questions, she constantly abuses in small ways and big, when she becomes passive aggressive, the psychopath and the narcissist reacts with extreme aggression. The aggression can, can have, can wear many forms. It could be silent treatment, could be verbal abuse, could be even physical abuse. At that stage, the borderline reacts with a primitive defense mechanism which is typical to two years old or one years old. And this primitive defense mechanism which survives only in certain mental health disorders such as borderline and such as narcissistic. So this defense mechanism is called splitting. Splitting is the common name. Today, today the clinical term is dichotomous thinking. We all come across splitting in immature or rigidly immature people. These are people who tend to divide the world into black and white. You're with me or against me. You're my enemy or my friend. You're all bad. You're all good. And they switch. They pendulate like a pendulum between these two positions. So the same person can be all bad yesterday and the most wonderful person tomorrow. So this is splitting. The minute the borderline perceives rejection and abandonment, she splits. And her intimate pub, the person she loved most, the focus of her universe, the pivot around which her life revolved, at that instant of perceived rejection and abandonment, at that instant, that very person becomes the embodiment of the reification of evil. He becomes a hated enemy. He becomes a villain. And so following splitting, the borderline is faced with a series of possible strategies. One strategy is to disengage, to detach, to move out. And many borderlines do that. They run away. Another strategy is to triangulate, to introduce a third party into the relationship in order to revive or resuscitate the positive emotions. Love, for example, via jealousy. So another strategy is to manage the relationship with jealousy. Not with intimacy, but with jealousy. So in this case, the borderline will ostentatiously flirt with another man. Or even upscore with another man, go away with him for the night. Or behaving ways which will provoke the partner into extreme feats of jealousy, or as the borderline would put it, would provoke the partner to notice her. Coupled with this, coupled with this transformation of the intimate partner into a persecutory object. And coupled with strategies, dysfunctional strategies, dysfunctional attachment strategies, such as running away, such as triangulation. Coupled with all this, there is an internal process of dysregulation. Pain, anger, paranoia, which is essentially a projection of the borderline's own aggression, are all internalized. And the borderline is overwhelmed by these emotions. She simply cannot survive these emotions. They are too much for her. She feels that she is drowning. She feels that she can't breathe. She feels that she is on the verge of doom. This imminent perception that she is about to utterly disintegrate into molecules, that she's about to die. It's the equivalent of emotional death. Dysregulation is one of the most horrible things to experience. And in the borderline, there is concurrent dysregulation of emotions and of moods. So she not only reacts with pain, anger, etc., but she also becomes dysphoric. She develops the equivalent of point-like clinical depression. And she feels that she absolutely must get away from the source of her frustration, pain and rage. She must get away from that source. And that source is, of course, the erstwhile intimate partner. She must dump him. She must get away from him physically. She must abandon him before he does this to her. She must preempt the abandonment. She must reject before she is rejected. At that point, her impulse control disintegrates. She no longer can control her impulses. In the best of times, borderlines cannot control impulses. They are so taken over by dysregulated emotions and moods that they don't have the energy needed to control urges and drives. And their decision-making process becomes irrational and very disrupted. It's a disruptive process. They, for example, cannot weigh properly the consequences of their actions. They don't have an analytic mindset which predicts the future and relates it directly to actions in the present. So in the best of times, there's a problem with impulse control. But when there is a question of rejection or abandonment or abuse, real or imagined, the dysregulation leads to extreme impulsivity which involves an externalization of aggression. The borderline's mental resources are insufficient, even when everything goes smoothly and well, for example, during the love bombing or honeymoon phases. But when things begin to go awry, the enormous amounts of pain and rage inside her can no longer be contained and they are externalized. Technically, she becomes a psychopath because psychopathy or antisocial personality disorder is an externalization disorder. So she, for example, could spot a stranger and decided he is her solution. She would immediately adopt that stranger. She would flirt with him aggressively.  And she would try to co-opt the stranger to collaborate with her in, for example, triangulation. If she chooses the triangulation strategy. If she wants to run away, she will pick up a small bag and just board the next bus to know her. Or the next train. She doesn't know where. So whatever it is, it's going to be impulsive. It's going to be improvised. She's not going to think it through. She just has to act because acting is her palliative measure. This is how she overcomes, she self-medicates against pain.  Let us continue with the scenario of the triangulation.  And frequently she will do it in the presence of the intimate partner as a way to kind of provoke him into jealousy, into noticing her, or even to punish him as an externalized form of aggression. But then, when she abscond with the stranger, when she is no longer with the intimate partner, there is a feature that is unique to narcissists and borderlines. And that is object inconsistency, object impermanence. Out of sight, out of mind, the minute she abandons the intimate partner, runs away, and aggresses, picks up a stranger for casual sex, whatever it is, the minute she is no longer physically with the intimate partner, the intimate partner is all but gone. She does not remember the intimate partner, nor does she recall or experience the emotions she used to have for the intimate partner. Actually, even in the best of times, many borderlines carry with them objects, mementos, trophies of the intimate partner, to remind them of the intimate partner when they are not with him in the same room. This is a very infantile and disorganized personality. Children have this, babies have this, object impermanence or object inconsistency. When mother leaves the room, as far as the baby is concerned, she is gone forever, she had never existed, and the baby starts crying, starts crying. So, it's the same with the borderline.  That's it, he had never existed. And all the emotions she had for him, and all the memories they share together, all the experiences they went through together, they are all erased. And when I say erased, I mean utterly erased, he never processes her mind for a split second. And so from that moment on, she feels alone, she feels independent, she feels free.  He had never existed. It's kind of rewriting of history, formatting the hard disk, and she starts from zero. She starts afresh, she starts new. With a borderline, every day, every minute, every hour is a new beginning because she doesn't have a continuous core identity. She has a series of self-states that interchange and replace each other. And so she is in flux. The borderline is a river. You can never step into the same river twice, and you can never encounter the same borderline twice. From that moment, for example, if she had found an alternative substitute for an intimate partner, another man, from that moment, that other man is her focus. He is her new source of internal stability. He regulates her internal functions. He is her new friend. And he is also her new protector. Because borderlines go through life, perceiving themselves as maidens in distress. They need rescuing. There's always a villain somewhere, usually the intimate partner. There's always a villain somewhere, and she's always looking for another man to rescue her, to protect her, to shield her, to support her, to provide her with succor, to sacrifice himself for her. And when the man she had picked up is grandiose when he is narcissistic, it caters to his need to be perceived as superior, as omnipotent, as godlike. And so they complement each other. Borderlines tend to pick up narcissistic men by assigning to them the role of protector and rescuer. As the time pass, the borderline develops Erzath's immediate intimacy with that new man. And the old intimate partner, only a few years ago, is long forgotten, had never existed, actually. She teams up with the other men, and she does everything with him, together. They had become a couple. This couple, this couplet, this diet, can survive for a few hours, can survive for a few days, or can become a new romanticism. That's in the example of triangulation. When she runs away, for example, it's the same thing. She looks to the new place or new location where she's going to. She looks to people she meets there, as their rescue ambience, I call it pathological narcissistic space, as the place where she would be safe from her intimate bond. It's all about the princess in the tower complex. She is in need of a knight in shining armor to kiss her on her forehead or lips, preferably, and to wake her up from this two-poor of abandonment, rejection and abuse which she had endured in her mind, if not in reality, with a previous intimate partner.  The stranger she had just picked, because she had converted him in her mind into a new intimate partner, a protector, a rescuer, and so on and so forth.  It's meaningful to the men, but it's rather meaningless to the borderline. The borderline tends to reward the men to gift him with her sex, because right now he's the focus of her world. She cares about him. She's attracted to him. She's grateful to him, her rescuer. She wants to serve him as his so-called slut to make him happy, to care about his well-being. It's all about the men. She gives the men her body. The men can service himself with her body because she needs protection. She needs to be rescued. Similarly, when she would go to a new environment, she would become extremely promiscuous. Her promiscuity is a way for her to engage with multiple potential rescuers and protectors until she homes in on one. And then, you know, there's a love-bombing phase and temporary attachment. The sex serves also to restore her self-esteem. She feels desire. She feels a center of attention, which is narcissistic. She feels wanted. She feels noticed. She feels de-abused. It takes away the effects of the abuse, the humiliation of the rejection, and the fear of abandonment. Now, her self-esteem is restored. She is validated. She feels strong. She feels determined. And now she can break up with a previous intimate partner if she bothers to do it. Very often they simply vanish. There's no closure, no breakup, no announcement, no nothing. Poof! They're gone. At that stage, however, there's a very complex inner process in the borderline. This borderline is premised, it's the outcome of childhood abuse, exactly like narcissism. But as opposed to narcissism, borderlines retain a modicum of empathy. This empathy is very functional empathy, but it's full-fledged empathy. It includes emotions, the emotional component. So borderlines have empathy, and as distinct from most narcissists, they also have negative emotions which are ego-dystonic. In other words, negative emotions which the borderline feels bed with. Narcissists have negative emotions, but they're happy about these emotions. Narcissists are angry, for example, so they externalize their anger antisocially, and they use it, they leverage it to get places. Narcissists are pretty happy, for example, they're envious, because they're envious, they're motivated to compete. So narcissists leverage their negative emotions in pro-social ways, actually. The borderline has negative emotions which are ego-dystonic. They are uncomfortable. For example, the borderline has fear. She has negative affectivity. She has guilt. She has shame. She has regret. Mostly, she's afraid to be caught in the act. If she cheats on her intimate partner, she's afraid to be caught cheating. She's very aware of what other people think of her, and in this sense, she's highly narcissistic. Remember that she self-regulates via the gays, or the combined gays of other people, exactly like the narcissists. The borderline has a hive mind, an outsourced mind. These two types outsource their minds because they have no internal environment. They don't have, their identity is diffused. They don't have a core. So the borderline is very worried. What would other people think of her if she's caught in the act? It may lead to retribution. She's very paranoid. Her image is tarnished. She also feels very shame and very guilty for having hurt her previous intimate partner, because she still retains some hope that maybe they could get back together. But if the borderlines had made a decision, an internal determination, that the previous intimate partner is not safe for her in the sense that he can reject her or abandon her again, forget about it. His gun, objecting constancy and splitting. He becomes all bad, and he's forgotten. Or if he's not forgotten, he's hated, and he becomes, he's transformed or transmogrified into a bad memory, an evil memory. So at this stage, the borderline would start to confabulate. Don't forget that the borderline, for example, during sex dissociates. She either depersonalizes, she feels that she's on autopilot, or that she's observing herself from the outside, or she derealizes, she feels that what's happening to her is a nightmare, a movie, not real. Or she becomes much more common amnesiac. It's called dissociative amnesia. She simply forgets things. She can forget whole stretches.  They dissociate them. They cut them with imaginary mental scissors, and they dump them in the wastebasket of memory. So there's a lot of dissociation going on. These gaps, this lost time, the borderline has to compensate for. And she compensates for these segments of lost time by confabulating. Confabulation is not like, it's not the same.  It's generating or creating an imaginary piece of fiction that is all the same plausible. It's like asking the question, what could I have been thinking? What did I do reasonably? What would, what did I do probably? What did I do, how did I act plausibly? And then the borderline says, well, okay, if that's what I had done probably, that's how I acted plausibly, probably that's what I did. And that's of course a form of magical thinking. What happens in your head is reality.  So that's confabulation. It's filling in the gaps, filling in the abyss of lost memory with plausible, probable, reasonable pieces of fiction. This is not the same like lying because lying is knowing what had happened and lying about it, altering it. It's a borderline is not aware of what had really happened. She is trying to second guess reality. Borderline, the borderline confabulations are egocintonic. In other words, she feels comfortable with them. She feels that they are really a part of her. She owns them. They become an integral patch within the fabric of her identity. Her confabulations are self-justifying. They are blame-shifting. So my fault. He made me do it. It's not my fault. He made me do it. This is called alloplastic defense. And borderlines have this a lot. They are never to blame. They never, you know, someone, circumstances made them do it. Intimate partner made them do it. Rejection and abuse and humiliation, perceived abandonment, all kinds of signs of cheating. They're very paranoid. They suspect their intimate partner of constantly cheating and constantly, you know, they're constantly jealous. And they have aggressive romantic jealousy. They externalize it and become very psychopathic about it. That's why we increasingly believe that borderlines are actually psychopaths, secondary psychopaths, not the male type of psychopathy, but a female type of psychopathy. But be it as it may, the borderline's confabulations are an integral part of her. Sometimes the confabulations are intended to restore the relationship with the previous intimate partner, to create a fluid narrative that will somehow accommodate everyone's needs. So she would say, yeah, I had sex with someone else, but I was drunk. You know, he got me drunk.  Or yeah, I went away for a few weeks because I needed my private time and space to realize how much I love you, this kind of thing. So these are confabulations. Confabulating is intended to breach the amnesiac gaps, but also to restore some normalcy to a situation that has been severely destabilizing, damaged and disrupted by impulsivity, aggression and defiance. And of course, the borderline, most borderlines abuse substances, many of them are alcoholics or drug abusers, junkies and so on. And so they would very often shift the blame to the drug or to the alcohol. So they would say, I forgot. I forgot.   I don't know what happened. This must have been a special type of weed, which made me do it. So they would shift. They would very often accuse the substances they had been abusing at the time they had misbehaved. They know they misbehaved. They're fully aware of it, but they can't own it because it's too disruptive. It's too ego-distonic. It will tear them apart. It will create, generate, because they are not substances. They feel guilt. They feel shame. They feel regret. And so if they own the misbehavior, they will be torn apart and they will be rendered asunder and they cannot accept this. Post facto, they would tend to try to revive relationships with intimate narcissists, with intimate partners. Even if the partners are narcissists, they'll try somehow to revive the relationship. They will keep in touch. Sometimes they would keep in touch in order to witness the intimate partner's pain and punishment. Sometimes they would keep in touch because they hope that whatever happened was a clarion call, a wake-up call. You know, now things will get better. Sometimes they are simply afraid because of their projected paranoia. They are afraid of the retribution of the intimate partner. They would keep tabs on the intimate partner. They would try to regulate the intimate partner's vindictive impulse by confibrillating in line so as to create a narrative where the intimate partner would not feel so hurt, would not feel so aggressive and hateful. And sometimes they keep in touch with the intimate partner after such a destructive episode because they resent the partner's moving, the partner moving on. They hate it that the partner gets over them and recovers. They try to pull him back. They try to tear him apart as well. Very often borderlines feel shame and regret what they had done. Very, very often. Many of them have auto-plastic defenses. They blame themselves. They say, you see, I'm so disordered and so disrupted. I'm so broken and I'm so damaged. It's hopeless. My life is hopeless. About 10% of borderlines attempt to commit suicide. There are certain sub-types of borderline. The figure is as high as 70%. For example, borderlines who are also diagnosed with dissociative identity disorder. 70% of them try to commit suicide. That's 7-0. So this indicates an inner struggle against shame and guilt and other egotistonic emotions. And many borderlines are actually good-hearted and kind-hearted and they try to provide closure. But they try to provide closure as they do all other things, dysfunctionally. So they would try to provide closure by lying, for example, or confabulating or providing soothing narratives. And this would backfire.  I mean, the intimate partner feels even more hurt and humiliated and offended against. And this massive deception and attempts to rewrite history and reality and so on and so forth. This is indicative of the borderlines in a landscape. It's a landscape of virtual alternative reality. Borderline has no real reality. You cannot have a reality if you don't have an identity. And you cannot have an identity if you don't have self-love. And you cannot have self-love if you don't have object constancy. And you cannot have object constancy if you constantly split. And all these are typical of borderlines and of narcissists. They just choose different solutions because borderlines are much more human than narcissists. We can recognize in a borderline our own struggles, our own pains, our own hurt, and how we try to cope with life's crisis and tribulations. When we look at the narcissists, we can't empathize or sympathize with the narcissists. Because there's nobody there. And in the worst case, the narcissist is robotic and mechanical and utterly frightening.