Warning about civil war in Russia. Former Russian commander Igor Ivanovich Strelkov, also known as Igor Girkin, recently warned of civil war in Russia that could result in millions of casualties as the country continues to fight in the ongoing war in Ukraine. There are all kinds of civil wars.  Girkin said in a clip with subtitles that was posted to Twitter by Anton Gerashchenko, advisor to the Minister of Internal Affairs of Ukraine. In the clip, Girkin added that Moscow could collapse into a civil war with millions of casualties, with a complete collapse and defragmentation. Russia continues to fight in Ukraine despite struggles in obtaining military equipment, shortage in manpower, poor morale among troops, and what some predicted an internal division within military leaders. Though Russian forces made some advances in capturing the Ukrainian city of Soledad, Ukrainian army leaders still have control over the situation in the western outskirts of the city, which is north of Bakhmut, Girkin said on Telegram. Meanwhile, warnings about a civil war breaking out in Moscow have been previously made by some of Russian President Vladimir Putin's critiques. Mark Fagin, a former deputy in the state Duma and Human Rights lawyer, told Newsweek last October that a comprehensive defeat for Putin in Ukraine could result in a bloody civil war with a range of factions and regions seeking power.


Today, as we close out 2016, I'd like to wish you all a very happy New Year's Day and I'd like to look back at what we've achieved in the past 12 months. I've traveled to every corner of Pennsylvania to listen to you and your neighbors about how state government can get back on track and make real progress for working class families. I have heard your voices loud and clear. We need to reform state government, make schools better for our kids, and create good paying jobs that support our families. So that's what I did. We're all tired of Harrisburg ignoring the working in middle class, which is why I've fought to increase wages and make investments in programs and projects that will create good family sustaining jobs. I've secured an additional $640 million in funding for education to restore the cuts made in the previous administration and have implemented a fair funding formula to make sure all of our children are getting the education they deserve no matter what their zip code is. I've also expanded healthcare access for nearly 700,000 Pennsylvanians and secured funding and bipartisan legislative reforms to fight the opioid and heroin crisis. We've legalized medical marijuana and ride sharing across Pennsylvania. We've reformed our liquor system for the first time since prohibition, making wine available in your grocery stores and six packs available at your local gas station to increase customer convenience. So today, I'd like to wish all Pennsylvanians a very happy New Year's Day and let you know that in 2017 and beyond, you can count on me to fight for you.


Good morning, everyone, and welcome to the sixth meeting of 2016 of the Public Petitions Committee and the final meeting of this session. I can remind all present, including members, that mobile phones and blackberries etc should be turned off as they can interfere with the sound system. Before going to our formal agenda, I would just like to bring the committee's attention to an issue in relation to petition 1593, which we have been considering over a period of time from fans against criminalisation. A complaint was made under the terms of the code of conduct section 7.3 volume 2 of the code details the requirements in relation to members' conduct during a committee meeting. This section of the code cross refers to rules 7.3.1 and 7.3.2 of standing orders. Section 9.1.6A of volume 2 of the code provides the complaints about the conduct of a member of a committee are to be referred to the convener of the committee. Upon receiving that complaint, I thought that it was only fair to afford the member complained about the opportunity to respond to the terms of that complaint. I therefore extended this opportunity to Kenny MacAskill as he was the person who had been complained about, but at no point made any comment about the substance of that complaint. This simply fulfills the responsibilities that come with the role of convener under the code of conduct, responsibilities that are determined by the Parliament and with which I must and have acted in accordance. I provided clarification to Kenny MacAskill of the basis on which the complaint fell to me to consider in that role and advised of how to go about seeking clarification from the standards clerk should he so wish. It appears to me that Mr MacAskill conflated an invitation to provide comments with a finding on the terms of the complaint, and he responded with comments that challenged both my authorities, the committee convener, and attacked me personally. I therefore asked him to withdraw his comments about my conduct and to apologise for them and to respond to the complaint that he has done neither. On November 21, 2012, I made comments in the chamber of the Parliament in response to which the Presiding Officer said that the standing orders of this Parliament explicitly state that members shall respect the authority of the Presiding Officer. It is inevitable that decisions of the chair will not meet the approval of all members at all times. That has always been the case. Nevertheless, it is imperative that the authority of the chair is respected at all times. I acknowledge that Mr MacMachan apologised. As Presiding Officer, I will support to the limits of my powers, the conduct of parliamentary business in this chamber. I will, however, not tolerate behaviour that falls short of the standards that the people of whom we are privileged to represent expect of us. The PO did that and found that the standing orders in the code of conduct—I have found that the code of conduct and the standing orders have offered me no such protection whatsoever in respect of the deliberate contempt with which Mr MacAskill has shown towards me and, more importantly, the petitioner and the third sector body who also complained about his comments. I just wanted to put that on the record because I feel that it is completely inadequate that the code of conduct does not provide any protection to the convener in this regard and to advise the committee that I will be writing to the standards committee to advise them of my concerns in that respect. With that, I will move on to the next agenda item, which is agenda item 1, consideration of continued petitions. The first of which is PE1319 by William Smith and Scott Robertson on improving youth football in Scotland. Members have a note from the clerks and the submissions and I welcome Chuck Brody to the meeting who has shown an on-going interest in this petition. Chuck, do you want to make any comments? If I am making any, thank you very much for allowing me to attend and to speak before I do so. I have just seen the letter from the SPFL and I may, with your indulgence, comment briefly on that. The answer reflects an answer that I received to the letter. I also wrote to the SFA in SPFL about the transfer and the payment of one individual that we have several others. I quote in the middle of paragraph 2. They say that we are often told of these payments, but to date we have been provided with no evidence and therefore are of great concern to us that there is continuing allegation and innuendo without any actual substantiation. I wrote to them highlighting the name of the individual, which I will not mention here, in the transfer between two Scotland senior clubs for a sum, we believe, in the region of £40,000 or £8,000 of which went to an agent. The body's SFA and SPFL said that they were not aware of that, and yet, lo and behold, on the front page of the website of the agent, they declared that they had registered the individual for whom we sought information. We will go back to them and we will highlight not just the one but the others that we have, but it is stretching it a bit far to say that they have been provided with no evidence. The agent has declared publicly that they have registered an individual, and I have to question the registration competence of the body's concern. In saying, I am delighted to speak. Before I really go into some substance, I will be brief. There are two real tests to this petition, which I was happy to speak to before and happy to be involved with those who brought the petition to us. One, in the most important aspect, as reflected in the Children's Commissioner's report, is the expectation, the expectation and the potential exploitation of children. Some, as I am told, are as young as six years old. The Children's Commissioner reports are invaluable to—I will show a window on what he believes is necessary and what he thinks has been happening. The second is the outcome for the performance of Scottish football generally. It is not good. Failure to an international level, the world level and the European level are evidence-based as to what the national performance is. It is far too centralised, too short-term looking at the future and skewed towards larger clubs. That does not help, as I see the expectation of young people who are brought into that environment. I believe that we need a major overhaul, a major reorganisation that has a stronger emphasis on Scottish youth and reflects Scottish youth. We need only, in my opinion, a national organisation that can do that. I have to ask which of the two organisations is truly fit for purpose with regard to the creation of a youth and elite academy that reflects the size of the Scottish population and does not try to emulate, as we have heard before, a disproportionate level of elite academy related to the population as we have tried to almost have equivalence with that of Germany. The overhaul in terms of young people's football, in terms of the role of the agents and qualification of them, particularly with regard to children and, of course, there is an emphasis on the role of parents. There should be clear guidelines and penalties on the role of agents. Indeed, consideration of the role of parents in how some might be seduced into being persuaded to sign their children up to professional football clubs. There is a wider role of clubs, I believe, in social and other local sports and community activities. Importantly, the last thing is to ensure that, in no way is there a default on the education of children who are involved, boys and girls who are involved, in trying to fulfil the expectation of being good professional footballers in some places, great professional footballers. That can only be done, I believe, if we return football meaningfully to schools, to communities, to boys and girls club at the grassroots, and to ensure that we begin again to believe that football should be competitive but should be fun for children that are not for sale. On that basis, I am more than happy to talk to you. Again, I applaud the petitioners for bringing it forward and I am happy to support the petition. We asked that the committee consider that the approval reaction be taken. Within the Parliament, I spoke with the Sports Minister several times about this, and I believe that the time for action is now. Thank you for your interesting contribution. Do members have any comments on Angus? Thanks to the petitioner, the SFA SPFL, I have taken the issues concerning youth football reasonably seriously. The petitioner himself has stated that the petitioner is moving the clubs and governing bodies in the right direction. We have also seen that the commissioner for children and young people has stated that it is clear that there have been some improvements in the approach that has been taken to children who are involved in youth football. However, there are still matters at standing. I also note the commissioner's comment that he no longer believes that matters that remain outstanding can be dealt with by self-regulation, and he recommends that the committee refers the matter to the Scottish Government to consider how the issues may be dealt with through external regulation, a suggestion that has also been backed by the petitioner. In light of those responses and taking on board Chick Brody's comments this morning, I would suggest that the committee should be minded to close the petition, but in doing so, it is right to the Scottish Government to ask for it to look at regulation through legislation to prohibit controls over young people that would not be permissible in any other walk of life, as the petitioner has already put it. Unlike my colleague Angus MacDonald, I do not think that we should be closing the petition down. I think that we should be, as Angus MacDonald has suggested, writing to the Scottish Government to seek its views regarding the regulation, because, clearly, the commissioner for young people has indicated that there are concerns in having sat through and listened to some of the evidence from the Scottish FA and SPFL. Clearly, they do not seem to be taking the issue as far as I am concerned as seriously as the general public and the petitioners. It is only by right now that we write to the Scottish Government seeking its views on it, but we include that as part of our legacy for the next committee. I think that there are still issues that need to be addressed here. By writing to the Scottish Government, I hope that the successor committee will get some response and some idea from the Scottish Government about what action it will take. Unfortunately, too often, we hear about self-regulation not being regulation at all. Organisations such as the Scottish FA and SPFL are not prepared to regulate or accept and acknowledge that there are some issues out there in relation to youth football, so we need to look at some form of regulation that captures and protects young people and their families from what could be seen as abuse from unscrupulous agents or clubs. I am minded to argue that we should keep the petition open as well into our legacy paper. The Scottish FA and SPFL are entities in their own right to any other commercial organisation. They have the right to make rules to govern themselves, but equally, they take public money. Recently, they have asked the Scottish Government to contribute to initiatives that they have been pursuing. I do not think that they can have it both ways. They cannot, on the one hand, make the argument that such cultural importance and importance to the Scottish Government has a responsibility towards them, but they do not have any responsibility to act in a way that those who represent the general public would consider to be appropriate. I do not think that they have been pushing hard against having a discussion around this. I think that they have indicated that they would be prepared to sit down in some sort of round table discussion. It could be that we could suggest to the Public Petitions Committee in the next session that they could pursue such a round table and see whether any discussions that could be had in a constructive manner could lead to some consensus being achieved as to how that could go forward. Mr Brody will be able to testify to the first attempt to get the SFA and SPFL to a committee meeting of the Public Petitions Committee. They resisted and felt that they were not accountable to the Scottish Parliament for their actions. The reality is that, although I agree, it might be useful to re-invite them to a future committee meeting, I advise any future committee to look back at the initial responses from both the SFA and SPFL in relation to the original invitation. It took some persuasion, as the best way to describe it, to get those two bodies sitting in a committee meeting to discuss and answer some of the issues that were being raised by the petitioners. It is just to make you aware of that, that they have given evidence in the past but reluctantly gave evidence. I, too, had meetings at Hamden Park with the SFA and SPFL. There are some things happening in Scottish football that are encouraging me. The fans' involvement in the role of hybs and hearts, for example, in widening the involvement in the franchise, are encouraging, but that is at the senior level. The meeting at Hamden again reiterated the need for evidence-based, which I think has been produced, but around the table is certainly a continuation of the petition with that information and with the involvement. I mean, I have had parents who have come to me with problems that they were having with the major clubs in Scotland and the impact that was having on children's education. If you like the binding contract, I will make contracts of inverted commas in terms of how the parents have signed up children to be involved and tied very much to one specific club, and hence the issue that we raised with transfer fees. I would be encouraged if the petition were to carry on as part of your legacy paper and that around the table, with more evidence than we had the first time, and perhaps bringing in parents who would be willing to comment on what actually happened in their experience. I thank everybody for the next committee to determine how to do it, but I think that we can certainly suggest that they consider such an action if we then keep it in the petition open. I take on board John's request to write to the Government and get a response in relation to that specific point, and I think that we can do both. Members agree? Angus, do you want to know? Just to say that it is a general consensus of the members to keep the petition open and to include it in the legacy paper, I am content with that, so long as John Wilson's suggestion about writing to the Scottish Government is included. With regard to a further evidence session, we have tried that in the past, however, a chick Brody's suggestion that parents should be part of that round table is certainly a good one if we are going to go down that route, because it is good to hear straight from the horsey smith, as it were. I reluctantly concur with the consensus to keep the petition open. I feel that this has been on-going since 2010, and I am not sure that we are going to make any significant further progress as a committee at all, as my own view. However, I note what the commissioner says, which is summed up in John Wilson's recommendation that we ask the Scottish Government whether it has any will to intervene, but it would be on that basis that I am agreeing with. I think that there is a consensus that we try to keep it open and pursue it a bit further. Okay. Thanks very much, Trichfer. Okay. Our next petition this morning is PE1412 by Billy McDowell on bonds of cation. Members have a note from the clerk and submissions in relation to this. I will open up to the committee to suggest what we do, if there is anything at all. I do not see any indication that there are any suggestions that we can bring forward. The next petition is PE1477 by Jamie Ray on behalf of Throat Cancer Foundation on gender-neutral HPV vaccination. I will invite contribution. I note the petitioner's request for the JCVI to accelerate its assessment so that the decision on vaccinating adolescent boys is made this year and not in 2017. However, I also note the JCVI's comments that it would be inadvisable to take shortcuts, which could undermine the validity of the results in order to expedite the review. While I can understand the petitioner's frustration at the length of time that this is taking, I think that we should take on board the comments from the JCVI. That said, convener, I think that we should write to the Scottish Government, as suggested by the petitioner, to ask that it considers early adoption of the programme and to keep this petition open as part of the legacy paper when we await the response from the Scottish Government. I would have been minded to recommend to close the petition on the basis that the objectives set out in the original request have been agreed to by the JCVI and that they will be making recommendations that will full the request of the petition. Therefore, I am not sure what further more the committee can achieve. I think that we could wait until we get the response from the Scottish Government. I take your point, Jack. There is not much left to do, but there is still a response due back from the Government if we write to them. I think that we could wait to see what that response is, notwithstanding the fact that I think that some progress has been made in terms of the JCVI. I would be minded to close the petition myself, but I am taking on board the petitioner's comments and I think that we should take it that way a bit extra. One more letter, I cannot do any of that. Agreed that we write and just keep it open for that response to come back. The next petition, PE1493, by Peter John Gordon on a Sunshine Act for Scotland. Members think about actions on this one. Stun, silence. Close the petition under rule 15.7 on the basis that the Scottish Government has undertaken to review the need for updated guidance on what the petition calls for and is consulting on the issue to gather views. Again, we have fulfilled the objectives that were originally sent out by the petitioner. It is not for us to do it, it is for the Government who has said that they will. Our next petition is PE1517, by Elaine Holmes and Oliver McElroy on behalf of the Scottish Mesh Survivors to hear our voice campaign on mesh medical devices. John Scott has joined us this morning and you have been pursuing this with some diligence. Do you want to make some comments before we consider the petition? I am very grateful for the Petitions Committee having carried out the work that they have. I am very grateful for the letter that has now been received from Cabinet Secretary Shona Robison in this regard. I think that the committee has taken this very serious matter forward in a very thoughtful and sensitive way. I am delighted that it has done so. I note from your paper that you believe that this petition should be extended into the next session of Parliament. That is a view that I would share if that is indeed the view of your other committee members. On behalf of the many affected petitioners, I congratulate you on your efforts so far, but keep up the good work, keep watching because I do not think that all the problems have yet been solved. I am very aware of recent reports in the press of huge claims having now been paid out in America with regard to injuries that have suffered as a result of these mesh implants. I can only see that that is likely to lead to more litigation here in this country. In terms of avoiding that in future, we have to be vigilant in how we look at dealing with this in problem in future. That is clearly a petition that we should keep open and include in our legacy paper, but I note two specific points in the cabinet secretary's letter. One is that we will be seeing a copy of the report on the single incision many slings. More urgently, the de facto determination that the regulation of medical devices is reserved to Westminster and that our concern about the conduct and interrogation of those devices by the MHRA cannot be addressed by our looking to the Scottish Government to set up any parallel or operation. Therefore, it will be useful for the committee to write to the UK Government minister, who is responsible for the MHRA, drawing their attention to the work that this committee has done on this subject and asking how we can influence the debate that it is taking place over whether or not those devices are being properly regulated by the MHRA over which we have such significant concern. I think that most people would agree that there is certainly more information appearing all the time about this. We are learning more and horrendous information about the impact of these implants. I think that there is a good bit of work still to be done on this petition and in general terms about the whole issue that is being raised. I think that there is a consensus on the committee that we keep this when open and allow the petitioner's committee in the next session to continue to scrutinise this. No-one has agreed. Our next petition is PE1548 by Beth Morrison on national guidance on restraint and seclusion in schools. I think that there is some work that we still need to do on this, and I think that writing to the Scotland's children and young people's commissioner inviting his views on the proposed guidance in the context of this petition and to include the petition in our legacy paper for the session 5 would be advisable. Members agree? Petition PE1563 by Doreen Goldie on behalf of Avonbridge and Standburn Community Councils on sewage sludge spreading. Thanks, convener. Obviously, I declare an interest as I have had a constituency interest in this issue. As we can see from the petitioner's submission, they have broadly welcomed and agree most of the recommendations that have come out of the Scottish Government's sludge review. However, saying that it has to be noted that the petitioners are disappointed that there is no planned reduction in the spreading of sewage sludge in the near future, I think that the petition has achieved a great deal. It has brought forward the review. In fact, you could argue that the review might not be happening had it not been for the petition. In addition, following pressure from myself and Margaret Mitchell, MSP and John Wilson, a large number of the points that have been raised have been taken on board by the Scottish Government. Coupled with the new legislation on the way and the knock-on consultation that will generate and the fact that the petitioners now have direct access to the lead Scottish Government officials, which again is a welcome development, I would normally have been minded to close the petition because it is a success story for the petitioners and for the petition's committee. However, despite welcoming the progress, I think that the committee should keep the petition open to allow us to keep a watching brief on any progress that is made with the introduction of legislation and also with regard to the issues that have been raised on giving SIPA more power to control the issues. I think that there is a strong argument to keep the petition open, convener, and I hope that other members of the committee are similarly minded. Thank you, convener. Like Angus MacDonald, I think that we should be keeping this petition open. I think that we have achieved a lot and there has been useful progress made. However, the petitioners have raised a number of questions on the submission on 1 March in response to the Scottish Government's action to date. When we are continuing the petition, it would be useful to write to the Scottish Government with the questions raised in the latest submission to find out whether those questions can be answered or dealt with as part of the review process to ensure that we do not just have a review in place and guidance issued, but we have the questions that are being asked particularly by the petitioners answered at the same time because there is no point in the Government coming out with a review document and saying that this is what we are going to do. Some of the questions that have been raised by the community council are still outstanding. It would be useful to write to the Scottish Government to say that there have been further questions asked, can we get clarification on those issues and ensure that they will be covered as part of the very guidance that is going to be issued, particularly the issue about CEPAS powers. As we have discussed in the committee before, it is okay to say that an agency has powers, but if the agency does not have the resources to carry out those powers, the powers will be effective when meeting us. I would have been minded to advocate that we closed the petition. It was a petition that called upon the Scottish Government to ban the use of sewage sludge. The Scottish Government has made it clear that it supports the practice of spreading raw sewage. I do not know if that is a more general metaphor, but in any event it has declined to support the aims of the petition. I hear what my two colleagues say, and if they believe that that would allow us to pursue the petition into the next session, I am minded to support that. However, it is pretty clear that the actual objective of the petition is not going to be agreed to by the Scottish Government. I think that we have been in that position before. We accept that the Government is not going to do what the petition asked for, but that does not mean that there is not an area of work that the committee could continue to scrutinise and ask questions around and hopefully get some answers in terms of the on-going review of it. As I understand it, the petitioners have submitted their own response direct to the Scottish Government. As I mentioned earlier, they now have direct access to the lead officials, which I am sure is welcomed by everybody. However, there is no harm in the committee submitting the questions that are raised so that we get some feedback direct from the Scottish Government. That is a good suggestion. I would be happy to take you open just to get that response back. At that point, the session 5 committee can determine whether there is any value in taking it beyond that. PE1577 by Rachel Wallace on adult cerebral palsy services. I think that we still have not pursued this to the fullest extent and there is a bit of work to be done. I do not think that there is anything that we can do specifically at the moment, but we cannot close the petition. I think that this is one that has to be left open. The meeting between Murdo Fraser, Ms Wallace and the minister was delayed until the beginning of March, which means that some of the work streams arising from that have not had a chance to mature, so I think that it would make sense to allow them to progress and for the committee to consider it in the next session. Petition PE1581 by Duncan Wright on saving Scotland's school libraries. I am very depressed by the response that we have received. I have to say that it almost to my mind validates the need for the petition, because essentially what many of the councils have said—I think that it is summed up very nicely here by Glasgow—is that school libraries are running different ways across the country. Given the financial challenges that councils are facing, that is an area where reforms have taken place, which essentially means closing libraries down or merging the more undermining and reducing facility. Most of the local authorities that have responded have more or less said, hands off, that is our territory. As far as I can see, they are undermining things. Last week we had what was it, a national good reader or whatever it was. I saw the first minister appearing with Harry Potter and Sundry and other characters, all reading books and encouraging children to read. That is not going to be possible if there are no libraries for them to go and borrow books from or educated and trained librarians who are in a position to advise on a system in that job. To my mind, there is a need for us to pursue that, not just now with local authorities but with the Government, because I do not see how that strategy can be fulfilled with the current way in which libraries in schools are being undermined. John Swinney, I totally agree. The petition was brought forward based on the on-going situation, but there was also a comment made by a number of people about the concerns about what would happen in this spending round with the budget restrictions that local authorities were going to face. I think that people thought that a bad situation was going to get worse. It would appear that that is exactly what is happening and I just do not think that we can walk away from the table with the shrug on our shoulders and ringing our hands. I think that we have to continue to press this issue, so I think that we have to keep this one open and let the future petitions committee continue to look into this issue based on what is coming out of the current cuts at local authority level. The next petition is PE 1595 by Alex Ander Taylor on a moratorium on shared space schemes. I just do not think that we have concluded our deliberations on this one. I think that there is a lot of information that we are gathering in in relation to this petition. I am certainly learning a lot about the concerns around it and I think that there is more that we can learn and we cannot close this petition until we have pursued this a lot further. I agree, convener, and again I was very disappointed by the quality of the response that we received and the variability of it. One council responded saying, However, due to acknowledged conflicting disability needs and preferences particularly surrounding car abuse, it is not possible to deliver a street design to meet the specific desires of all disability groups. I just thought that that does not really take matters very much further forward. Given that what was being represented to us is that the street schemes that were being introduced were actually more prejudicial to disability groups than they were before they became shared schemes, I do feel that this is an area where there is a need for further action. I also note the comment by Lord Holmes that the UK Government has asked for all UK authorities to submit details of any shared space schemes that they currently operate in order that a review can be undertaken. I do think that if the UK Government is pressing a review of the way in which those schemes are functioning, there may be a case for a parallel initiative here in Scotland. I would make one comment in terms of some of the submissions that we have had. There appears to be a belief or a misconception that this committee has got the authority to order local authorities not to do those things. I just think that we have to put on record that we do not have that authority, but what we can do is scrutinise what is happening and ask for policy direction to be looked at. However, we cannot make a ruling in terms of the ability of any local authority to implement such a scheme or otherwise, and we certainly cannot order a planning authority to undo any decision that it may have made. John Swinney? That perception may arise from the East and Barnasher submission where they described this committee as the Scottish Government Public Petitions Committee, if only. We are the Scottish Parliament's Public Petitions Committee and we do not have legislative powers to force the Government to do that. I just think that we have to make that clear in case people are working under a misapprehension. We will definitely keep that petition open and take it further forward. Next petition, PE1596, by Paul Anderson, James McDermott and Chris Daley, on the in-care survivor service Scotland. Given the on-going discussions that we are aware of taking place, I do not think that we can close this petition. We have to leave this in our legacy paper. Members agree? Our next petition is PE1597, by Bill Welsh, on Michael Plasma fermentans in regressive autism. We really have to close this petition and the Scottish Government, which acts upon the advice that it received, has made it clear that it does not support the aims of the petition and therefore I think that it is impossible that it can proceed. Members agree that that is the case? That brings us to agenda item 2, the annual report. It is consideration of the committee's draft annual report for the parliamentary year 11 May 2015 to 23 March 2016. All committee annual reports follow a standard format, as agreed by the convener's group, members have a note by the clerk and the annual draft report. Do you have any comments on it? It is a pretty standard document. Everyone is happy to agree the report then. That is agreed. As agreed at our last meeting, the committee will now go into private session for item 3 on today's agenda.


I think that people need to understand comfort zones and why they exist and how to get rid of them. So, like we were talking about earlier, thousands of years ago, we had to be afraid. We had to have fears. We had to have worry in order to keep us alive. We're inside of this cave, and we know that we're alive inside of this cave. But we don't know if we step outside of this cave that we don't know we're going to still be alive. There could be a lion in the bush that could attack us. And so what happens is, if we were going to walk out of this cave, we would get a full physical fear in our body saying, no, we probably should not do this. And we would get that physical fear. Now, we don't have to have that physical fear anymore because there's no lions or anything like that outside that are going to attack us, most of us at least. So what we do is we fill our brains with other fears, surface level stuff, what people think of us, being rejected, fear of success, all of those things. And we know that the comfort zone that we're in, our brain that wants to keep us alive, knows that the comfort zone that we're in, that we've been in for six months or years or 20 years or some people 30 years, they've been in the exact same spot.  Because they get to this comfort zone, they stay there forever. And the reason why is because we know in this comfort zone, our brain knows that we're alive. It knows that we are not going to die because we're in this comfort zone and it has data that where we are now is alive. But what we've done to get here, we're still alive.   It also doesn't know that you get outside of this comfort zone of go and make $100,000 that you're not going to die on the way to getting there.   And so once you stay inside of this cave, just like we were cavemen, it doesn't want you to get outside of it. It doesn't want you to get outside of this comfort zone. We have to push yourself or work harder or work on yourself more because it doesn't know, you know consciously you're not going to die. But it doesn't know in that little teeny part of our brain, animalistic reptilian brain that we have, that we're not going to die when we leave that comfort zone. And so I think that if people can understand how their brain works and why comfort zones exist, they can sit there and go, okay, I have this video. Like I take my camera and I remember when I was going to post my first few videos, I was like, what are people going to say? Like why the imposter syndrome, right? Like I'm not as good as Tony Robbins. Why would somebody listen to me when there's Tony Robbins out there? And so there's that physical fear of like, I'm about to post my first video. I'm about to put my first podcast, my first picture, whatever it is. What are people going to say? What are people going to think? And there's that fear because I've never done it before. The physical, the butterflies and like your arm hairs kind of stand up where it's like, I don't know what's going to happen. But I could sit in that moment and go, why am I feeling this way? I think that people can understand why a comfort zone exists is to keep us alive and realize that most of the time we push ourselves out of our comfort zone, we're never going to die, then things really start to change.  So a rubber band, you pull it and it goes back to the same size. Plastic bag, if you pull it, it will never go back to the same size. It will now be a little bigger than it was.   And now you have all of the stuff that you could do that you're more confident in because you came a little bit out of that cave, because you came a little bit more. And so I think if you can understand how your comfort zone works, how it exists, why it exists in our brains, you can look back as a thinker and say, why am I feeling this way? Do I like it? I don't. All right, I'm going to do it anyways.


In the related development, the Sultan of Saqr to Al-Haji Mohammed Sa'ad al-Bubuqad III says the killing of worshipers by the government of Saint Francis Catholic Church is condemnable. According to the Sultan, it is mandatory for security agencies to help in rescuing the helpless citizens that all Nigerians have become. The Sultan described Heila Sitaq as a crime against humanity and a grievous and inexplicable sin. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Authoritarianism keeps surging in Western free democracies. Today in tyranny we've got three stories on the rapidly increasing authoritarian abuses in Western free democracies. Let's dig in. Number one, Greyzone reporter detained by British counterterrorism police for doing journalism. The Greyzone's kit, Clarenburg, was detained by six anonymous plainclothes counterterror officers who grilled him for over five hours about his reporting upon returning to Britain on the 17th of May, according to a new report by Greyzone editor Max Blumenthal. Blumenthal reports that Clarenburg was asked many questions about the Greyzone and his work with the independent outlet, saying police seized the journalists' electronic devices and SD cards, fingerprinted him, took DNA swabs, and photographed him intensively, threatening him with arrest if he didn't comply. Blumenthal writes that the police action was likely a retaliation for Clarenburg's reporting for the outlet, which has angered British officials and establishment media figures with the inconvenient information it has reported about their behavior. Here's a quote. Clarenburg's interrogation appears to be London's way of retaliating for the journalist's blockbuster reports exposing major British and US intelligence intrigues. In the past year alone, Clarenburg revealed how a cabal of Tory national security hardliner has violated the Official Secrets Act to exploit Brexit and install Boris Johnson as Prime Minister. In October 2022, he earned international headlines with his exposé of British plans to bomb the Kerch Bridge, connecting Crimea to the Russian Federation. Then came his report on the CIA's recruitment of two 9-11 hijackers this April, a viral sensation that generated massive social media attention. Among Clarenburg's most consequential exposés was his June 2022 report unmasking British journalist Paul Mason as a UK security state collaborator hell-bent on destroying the grey zone and other media outlets, academics and activists critical of NATO's role in Ukraine. Asserting that Clarenburg did nothing more nefarious than engaging in the same journalistic practice that the West's most prominent legacy newspapers from the New York Times to the Washington Post depend on to break news themselves, Blumenthal says it appears that British authorities did not detain Clarenburg for any legal breaches, but because he reported factual stories that expose the national security state's own violations of both domestic and international law, as well as the maligned plots of its media lackeys. Blumenthal himself was subjected to legal harassment and intimidation in the United States a few years back, arrested and charged with having committed assault while reporting on imperial efforts to drive the Venezuelan government out of its embassy in Washington DC. The charges were later dropped. The grey zone has been doing some of the best independent reporting in alternative media over the last few years and should wear its now evident status as a thorn in the empire's side with pride. Number two, South Australia passes draconian anti-protest law. Reacting to recent inconvenient demonstrations by environmental activists, the state of South Australia has just rapidly shoved through legislation without consulting the public to exponentially increase the penalties for unauthorized protesting. Demonstrators will now face up to three months in jail and fines of $50,000 if they are deemed guilty of the extremely vague offense of obstructing a public place with their protesting. Human Rights Law Center expresses the following, quote, South Australia is the latest jurisdiction to impose severe penalties on people for engaging in peaceful protest, joining New South Wales, Tasmania, Victoria and Queensland, who have passed anti-protest laws in the past five years. South Australia's anti-process laws carry the harshest financial penalties in Australia. The bill is excessive and will have a chilling effect on the right to protest in South Australia. The bill is also potentially unconstitutional and in clear breach of well-established principles of international human rights law, end quote. South Australian Premier Peter Melanoskas acted shocked and offended that anyone could possibly think life-altering penalties for vaguely defined protest activities might have some effect on protest activities, saying, One of the things that I have found rather disconcerting about some of the commentary on this piece of legislation is that somehow it curtailes or diminishes people's right to protest, which is simply not true. Hilarious. Now would probably be a good time to repeat my periodic reminder that Australia is the only so-called democracy in the world, which has no national charter or bill of rights of any kind. A lot of attention went into the Australian government's authoritarianism when its strict COVID measures were in place, but the fact of the matter is that this country has been diving headlong into tyranny since long before COVID and continues to do so now that the lockdowns are long over. There simply aren't enough checks and balances in place to prevent this from happening, and not enough will from the public to fight them while fighting is still possible. Number three, State Department dismisses questions about Ukrainian imprisonment of US citizen for speech crimes. At a press conference last week, the State Department's new spokesman, Matthew Miller, flatly brushed off questions about whether the US government was doing anything about the fact that commentator Gonzalo Lira has been arrested and charged with what amount to speech crimes by the Ukrainian government. Here's the State Department's transcript of the exchange. It's a question from Liam Cosgrove with Epoch Times. He says, a US citizen who is residing in Ukraine has been arrested. He was a California-born man. He was in the past like a business insider contributor, and he had a YouTube channel. He was an outspoken critic of Zelensky's regime. The Ukrainian SBU released a press release saying he was arrested for justifying Putin's invasion. So ultimately, it added up to speech. And I spoke with Congressman Ted Liu, a Democrat, and he said he urges the State Department to engage its authorities to work out some sort of negotiation to get him released. So are you guys aware of this? How do you feel about our allies detaining US citizens for speech abroad? Miller answers. So I will say in general that we're aware of the report. We obviously support the exercise of freedom of speech anywhere in the world, and I'll leave it at that. And the question comes back. So you guys aren't working to get him released? Miller answers. I'm going to leave my comments where I just left them. It's not every day a US spokesperson gets asked a question that's so inconvenient that they just overtly refuse to answer it without even pretending to provide an explanation for doing so. Lyra, a US citizen, is reportedly facing five to eight years in prison for having, quote, publicly justified the armed aggression of the Russian Federation and publicly justified the armed aggression of the Russian Federation per the SBU. Are Americans OK with their government risking a very fast, very radioactive Third World War to defend the freedom and democracy of a nation that imprisons US citizens for speech crimes? I guess we'll never know because nobody's asking them. If Western governments need to keep ramping up censorship, propaganda, and the persecution of journalists in order to defend Western freedom and democracy, is it really freedom and democracy? And is it worth defending?


Today and in the next few lectures we are going to study a very important class of continuous Markov processes called diffusion processes and there is a very large body of literature on these processes and very large number of applications both in physics and in other subjects chemical physics for instance chemistry and so on and so forth. But what we are going to focus on is not so much the detailed technical mathematical rigorously mathematical aspects of the subject as the possible applications to various physical situations. Let me begin by recalling to you that we had just started defining continuous Markov processes and in particular I said we will talk about stationary continuous Markov processes we can relax this assumption of stationarity a little bit and talk about stationary or non-stationary processes and as you will see as we go along the most important non-stationary random process is in fact the diffusion of a particle that itself its position is a non-stationary process the velocity turns out to be stationary but the position is non-stationary as you will see when we go along. So to start with recall that we said that these processes are described by the conditional density which satisfies an equation of this kind in the stationary case X is the random variable and a set of values taken by the random variable and the probability density function the conditional density function conditioned on this initial value satisfies an equation of the form an integral over all possible X primes dx primes and then inside you have a W of X X prime P of X prime t at X naught minus the loss term which is W of X prime X P of X t this is the continuum analog of the discrete equation that we wrote down in the case of a process which took on a discrete set of values or attained the discrete set of states. Now as in the other case this is an integral differential equation although it is linear in this P and therefore is not very trivial equation to solve certainly if it had been a matrix equation we could have written the solution as the exponential of a matrix multiplied by t and then try to look for methods of exponentiating this matrix but here that is not true this is some kernel so it is an integral differential equation this is some kernel function of X and X prime d t o here and it is not at all so obvious what the solution is here one approach would be to try to convert this to a differential equation but because it is an integral equation and you are integrating over all values of this here and no conditions have been put on these transition probabilities at all these transition rates it is immediately sort of intuitively clear that the order of the differential equation in the X variable will tend to become infinite in this case and in fact that is so and I will write that down without going through the intermediate steps except to indicate how to do it what you have to do is to treat this as a function of X prime and you put X for instance is X minus X prime you put it equal to some delta X or something like that does not have to be small and then you do a Taylor expansion in terms of this delta X and whenever you get derivative operator is you try to put it on the P by integration by parts and the result is that this becomes equal to also equal to and I am going to skip these steps as summation from n equal to 1 to infinity minus 1 to the n over n factorial delta n over delta X n a n of X P of X t for given X naught of course. So it becomes equal to that formally these two are equal to each other where these coefficients a n of X equal to an integral of moments of this guy X plus delta X starting with X for example and then delta X to the power n and D delta X this difference and the nth moment of this increment with this as the weight factor is the definition of a n of X here and delta X is over the range allowed range here so it is a definite integral this thing is a definite integral and it is a function of where you start namely X. So this is the exact formal equivalence now of course it immediately raises the question of when is this valid when is it convergent and so on and so forth I am not going to talk about those technical issues at the moment except to say that you can make specific conditions put specific conditions under which this equation reduces to this infinite order partial differential equation and this is called the Kramers-Moyal expansion does not serve much purpose except for formal purposes because it is an infinite order differential equation. So it is not any easier to solve than this but it gives you a little bit of physical insight as to what are the terms that are contributing out here and how do you interpret them and so on and so forth okay but there is one great simplification that occurs for a specific class of processes called diffusion processes and by this I do not mean I mean diffusion in a technical sense which is not restricted to the physical diffusion of a particle in space or anything like that but a mathematical term which says there is a class of processes called diffusion processes for which this equation simplifies enormously okay and there is a theorem a rigorous theorem which says remarkably enough if these moments of this increment delta x the amount by which x jumps to a new value if these moments vanish for any n greater than equal to 3 so happens that a n is identically 0 for any n greater than equal to 3 then a n is equal to 0 for all n greater than equal to 3 it is called Pavoula's theorem and it is a remarkable theorem it is not magic it is possible to derive it very very fairly straightforwardly but the statement is if a n equal to 0 for some n greater than equal to 3 and of course that immediately simplifies matters enormously and processes for which this happens are called diffusion processes because that would immediately imply so that this equation becomes the following delta p over delta let us leave out all the arguments of this p equal to minus delta over delta x a 1 of x p plus 1 half delta 2 over delta x 2 a 2 of x because the higher moments are 0 and this 2 factorial I have put in here as a half there this is called the Fokker-Planck equation they originally derived it in a different context altogether in a related context but in a physical context of a particle diffusing in space and this is was the velocity of this particle but today we call it the Fokker-Planck equation in general for any diffusion process and I will use the same terminology and you can go ahead and interpret what this means and it will turn out we will see the specific examples that this term represents the effect of noise on this variable x that is what causes x to fluctuate randomly whereas this term very often describes the effect of deterministic evolution in this x as we will see from the examples so this part is what x would do its distribution would do in the absence of any noise and this is what makes it random so very often this is called the drift term and that is called the diffusion term and we use we will use these terms in general even though they come from the physical application I am going to talk about so this is if you like this is the drift and this portion is the diffusion it is still not a trivial equation as you can see because it is got this term here it is a second order in the position variable I will frequently call this the spatial variables because for want of a better term although it need not be that at all in fact in the original context of the Fokker-Planck equation it was a velocity variable it is first order in time but second order in the other variable so this is technically not as simple an equation as say Laplace's equation or Poisson's equation because of this inhomogeneity you will recognize special cases of it for example if this A1 had not been present and if that A2 were a constant this looks like the diffusion equation the ordinary diffusion equation for particles diffusing on a line where which would be delta P over delta T is D times D2 P over DX2 we will see how that comes about now of course one could ask what happens can I solve this equation in general and so on the answer is no in general for arbitrary coefficients A1 and A2 it is not so trivial to solve at all can I incorporate non-stationary process in this yes indeed that has nothing to do with the vanishing of the moments or anything like that it is an independent statement if these were time dependent explicitly then of course you have a non-stationary Markov process and then you have to be careful you have to write X0, T0 etc keep track of that and then this would become time dependent here and it would be time dependent here because W would be time dependent explicitly so it is possible to incorporate non-stationarity into this business by looking at time dependent coefficients we are not going to do that all the cases we look at would be stationary in that sense but we will come across a non-stationary process we will see what happens in that case where you do not have explicit time dependence and yet you will have a non-stationary random process which will be the position for instance we will see how it comes about okay. Now to make it familiar with the diffusion equation one possibility is to derive this diffusion equation independently all together we already looked at diffusion on a linear lattice in the presence of a bias we looked at a random walk and the question was can the random walk be made into an equation of this kind then of course you begin to see immediately the connection between these two so we will do that now but one thing I want to point out is that there is a special case of even this Fokker-Planck equation and that is very very important it is a very basic process in fact it is the most basic Gaussian stationary Markov process it is just one of them it turns out and everything else can be mapped on to that and it is called the Onstein-Volenbeck process it corresponds to the case in which you have the following extra simplification so the case in which a1 of x equal to a1 x and a2 of x equal to a2 a1 to equal to constants so a case in which the drift is linear in x and the diffusion term is just a constant this coefficient a2 is a constant this is called this particular process then is called an Onstein-Volenbeck process so let us write it down a very important special case and we will spend some time solving this the density for the density function of this Onstein-Volenbeck process along with the physical example but in the meantime let us go back and see whether we can derive this kind of equation from the random walk model altogether so let us go back to the case of a biased random walk on a linear lattice in one dimension so if you go back and recall what the statement of this problem was we had a linear lattice and infinite lattice say label by the site index J some arbitrary site was the origin and then you had a probability if you are at the site J of jumping to the right with the probability alpha and to the left with the probability beta and this was true at every site you would toss this unfair coin and you either jump to the right or to the left. Now we did that in the discrete time case but we also did it in the continuum we said the time was continuous and the steps were given by a Poisson process with some mean rate lambda in which case the process that corresponded to right steps steps to the right had a mean rate lambda alpha and those to the left had mean rate lambda beta and if you recall the master equation in that case was dp J, t I suppressed the fact that we started the origin we keep that going so that just a matter of simplifying the notation this was equal to lambda times alpha p of J minus 1 t plus beta times p of J plus 1 t minus alpha plus beta p of J comma alpha plus beta is equal to 1 but I put that back here as alpha plus beta because I am going to recombine terms so do you recall this this was the master equation for this probability p of J comma t and the initial condition was p of J comma 0 is delta of J comma 0 the Kronecker delta we solve this and we discovered the distribution was a modified Vessel function ij of 2 lambda t square root of alpha beta etc right now we are not interested in that solution but we want to see what happens in the continuum limit when this J becomes a continuous index so what we do is to introduce a lattice constant this spacing a I am going to let a go to 0 and correspondingly let the rate of jumps lambda become infinite because the distance you have to jump is going to go to 0 and the rate becomes infinite in a specific manner so as to derive a finite limit on for the right hand side a proper limit for the right hand side so the first step is to write this as equal to lambda times let us choose this first beta of p J plus 1 t minus p of J t minus let us subtract the other difference also p of J t minus p of J minus 1 t so I have taken care of this term and this beta here in this portion here and then I added I subtracted that too so I need to add that back so this becomes plus lambda times beta minus alpha times p of J comma t minus p of J minus 1 so that is the other term did we go through this continuum approximation earlier have we explicitly done that okay so it is worth looking at it carefully to see what exactly is involved so what I have done is to add and subtract this thing here and I get this now it is clear what you should do in order to get the continuum limit because this looks like the second difference this looks like p J plus 1 minus twice p J minus p J plus p plus p J minus 1 so this looks like the double difference the second derivative and this looks like the first derivative if J were a continuous variable right so what we need to do is to multiply and divide by the lattice constant and take limits so out here I need I can rewrite this this thing can be rewritten as p of J plus 1 t minus twice p of J t plus p of J plus 1 t I divide the whole thing by a squared because it is a double difference here and multiply by a squared I multiply this by a and divide this by a and take the limit so the correct limit that we need to take the continuum limit is lambda tending to z infinity a tending to 0 and I want lambda a squared to become finite that can only happen if beta minus alpha also tends to 0 so that I get something which goes like an a squared here so we need a times such that and J tending to infinity because what I am going to do is to put J a tends to x so J also becomes infinite such that J times a is my x coordinate just like we went to the continuum limit in time by saying the time step n multiplied by the by tau the unit the unit time step was such that n tends to infinity tau goes to 0 such that n tau went to t the continuous variable exactly the same way J tends to infinity a tends to 0 such that J goes to the variable x such that what we need here is lambda a squared beta limit lambda a squared beta is finite equal to some number D by the way if alpha tends to beta this is the same as the limit half lambda a squared equal to D because beta is also going to go to alpha and a times alpha minus beta times lambda tends to limit equal to what would be the physical dimensions of this limit of this quantity that is a length and that is a rate velocity just a velocity so let us call it alpha minus beta equal to C then then this quantity P of J, t tends to the probability density P of x, t but you got to pay attention to the fact that there is a dimensional change here because this is dimensionless probability that is a density probability density it has dimensions 1 over a 1 over length right so you have to be careful about it there is an extra a factor there which you can put in but it is not so serious because it will appear on both sides out here and when you take that limit you end up with this equation becoming delta P over delta t of x, t starting from some x not you do not care we do not put it here equal to this quantity is D and now here here we have minus C delta P over delta x because that is the first derivative plus D and that is exactly in the form of this Fokker-Planck equation that we have written down so in this problem a 1 of x equal to C a 2 of x equal to D both of which are constants a very trivial example in which these coefficients have actually become constants here so the position in the case of bias diffusion the position variable is a see looks like it is the Markov process it obeys a Fokker-Planck equation with constant coefficients both the drift and the diffusion terms are constant looks exactly like that right this kind of equation for the positional probability density when you have diffusion in the presence of an external field this fellow here looks like it is a drift caused by some external field because you are saying systematically either alpha is bigger than beta or smaller than beta it drifts to one side whichever is larger and that is exactly what happens when you have a constant force on the particle so this looks like this equation is describing the diffusion of a position of a particle positional probability density of a particle subject to diffusion but under a constant external force of some kind and indeed it is so it is indeed so because if you recall the problem of sedimentation that we talked about this is exactly what happened you had an extra term exactly of this kind we even saw the solution we wrote the steady state solution I think in that case and what did we get in that case the problem we looked at was I said J equal to 0 here 1 here 2 here and we looked at a case where this part was bounded and then it turned out that P of J P stationary of J was proportional to the bias alpha or beta so you have bias such that these rates are alpha and these rates are better downwards and this was proportional to alpha or beta to the power J this is what we had we imposed a boundary condition on this we said it cannot go below 0 so the rate alpha minus 1 to J equal to 0 was 0 and the rate from J to minus 1 the beta 0 was also equal to 0 then we immediately got this as a steady state solution and we need to normalize this we need to normalize this guy over J from 0 to infinity should be equal to 1 and that of course you sum this geometric series is 1 over 1 minus alpha over beta which is beta over alpha minus beta correct so this whole thing is proportional to this guy so equal to whatever is normalization in this problem beta was greater than alpha this is the steady state solution we got but we will now let us try and take the continuum limit of it here and see what you want to get so I need to put all these guys in I need to put in all these fellows here back again so let us do that alpha is going to go to beta but I can write this fellow here in this problem beta is bigger than alpha in this case so we got to be a little careful here about the sign I define my drift velocity C as alpha minus beta so if C is positive it says alpha is bigger than beta otherwise C is going to be negative we have to remember that sign here so let us write this as e to the power J log alpha over beta or log beta over alpha with a minus sign and I am going to take the limit in which alpha is equal to beta so this becomes the limit that guy becomes one the log of one it is going to get so what should I write I write this as log beta minus alpha over alpha plus alpha I can write it like that surely which is one plus beta minus alpha over alpha I mean we can do this very rigorously but you can see what is happening and this is going to go to 0 beta minus alpha what is log 1 plus e as e goes to 0 the leading term the itself right so this is can be replaced as beta minus alpha by alpha this form apart from some normalization we will worry about that later and I want to make this J into x so I multiply by an a divide by an a but a times beta minus alpha is going to C right so let us multiply this by another a this is what I had and I multiply this by another a so I got to put another a here and let us put a lambda here and a lambda here we all said to take this limit because what does this whole thing go to this fellow goes to x that guy goes to C minus C and this fellow goes to D where did the x come from alpha this was an alpha which is the same as beta in the limit right so you are going to get something like e to the power minus e to the power C x over D where C is negative I probably use the symbol C for the downward velocity limiting velocity but I have defined this C as an upward positive in the upward direction so increasing J so that is why I change the sign here but this is exactly what we got earlier we interpreted this as the Peclet number and so on but that is exactly what this gives you this equation gives you because you go back to this equation and ask what is the stationary solution what is it going to be P stationary satisfies the equation D D to P over DX to stationary minus C D P stationary over DX equal to zero that is what this tells you but I can pull out a D D over DX from here and write this as D over DX DP stationary over DX minus C times P or C over D so equal to zero and C by the way is minus this is plus modulus that is the equation that is when the what is the solution what is the solution to this equation not quite not quite because what you can what you can say from this is that this quantity is independent of X so this guy must be equal to some constant independent of X whereas here we did not have any such problem we did the random walk problem and we immediately got the answer right away but here we are getting an equation which says this follows actually independent of X nothing more than that what would you have to do to match that to this and make sure that that constant is zero you have to put a boundary condition somewhere we already put a boundary condition on the floor we said it cannot go below the floor we already did that here in this case we have not yet imposed that condition there we need to impose that condition which will be precisely that this quantity is zero at X equal to zero because this is the flux remember that this equation can be written in the form of an equation of continuity because I can write this as equal to minus delta J over delta X where J of X equal to D times D P over DX in this case plus mod C P so it is in the form of a continuity equation in this case and that is the flux at any point because it is precisely a continuity equation for this probability density and then it says you cannot go through the floor so it means this quantity this current here must vanish at X equal to zero but in the stationary case and only in the stationary case this quantity is independent of X completely and since it vanishes at X equal to zero it must vanish for all X because it is independent of X okay so I emphasize again this quantity is not zero for X not equal to zero in general there is a current otherwise you would have you would not have any dynamics at all certainly P of X comma T in general is a function of T okay but when you go to the stationary distribution there is no T dependence anymore okay so the statement is that the boundary condition says that the current as a function of T vanishes at X equal to zero for all T what a partial differential equation I have to give you an initial condition and I have to give you a boundary condition the boundary condition must be valid for all T the initial condition is valid for all X for a given T right at T equal to zero so in this case this acts as a boundary condition and it says this quantity here vanishes at X equal to zero and that same boundary condition applies even in the stationary distribution but in the stationary distribution you discover that this quantity must be independent of X and since it is zero at X equal to zero it is zero everywhere identically and once you put that in this is the solution so you see our discrete model went exactly into that that is just a verification that these limits are all right that all these factors were right just right and it gives you this equation here in this special case in which you have this particle diffusing under a constant force field here you can apply to other cases it could be an electric field causing a steady drift or whatever but this is the exact continuum limit so the lesson is that the bias random walk with a constant bias the same bias at all sites is equivalent to the diffusion of a particle under a constant force field in the continuum limit but now we are approaching the whole thing from the continuous Markov process angle okay so we are going to write down although we did not have any differential equation for the position of the particle in the random walk problem but only difference equations for the probability density now that we have a continuous Markov process we could go back and ask one more thing which is to ask okay it is a random variable but does the variable itself satisfy a differential equation or not this is not a differential equation for the variable it is a differential equation for the probability density of this variable and that is a nice object but the variable itself will be very irregular will be random because it is being driven by some fluctuations in this case we are going to find out that this will satisfy a differential equation but it is what is called a stochastic differential equation a random differential equation and it should correspond and be consistent with the fact that the probability density satisfies this master equation here the general name for particle which is for the positional probability density of a particle under diffusing under an external force field it is called a Smolukowski equation so this is an example of the Smolukowski equation I will call it a Smolukowski equation because it is much more general than this you already saw that this has the effect of constant force field what would you say happens would happen if there was a force here explicit force which was position dependent how do you think this equation would change it would not be a constant this a 1 of x would not be a constant right a 1 of x in some sense would be the force would involve the force so if the force were due to a potential v of x I would expect that something like minus v prime of x appears in this drift term okay and then the diffusion the scattering would come from the d part here so this is something to keep in mind that the first term will be a drift due to deterministic forces and the second term would be the diffusion due to random forces okay I will will systematize that so let us go back to this Fokker-Planck equation and ask is there a correspondence between an equation of the form delta p over delta t I will continue to use the variable x which does not have necessarily the connotation of a position but a random continuous random variable Markov process so delta p over delta t equal to minus delta over delta x a 1 of x t plus 1 half this is the Fokker-Planck equation I turns out that this Fokker-Planck equation is entirely equivalent to a certain stochastic differential equation for this variable x random variable x which is now called a diffusion process in the mathematical sense and that equation is the following I will write it down but I am not going to prove this that this is entirely equivalent to a certain differential equation for x itself which reads in sort of physics notation it is not the most rigorous notation it reads x dot equal to some function of x perhaps even a function of t if this is a function of t but we are looking at stationary processes so let us just call it f of x times plus g of x times a white noise and let me call it by eta t and explain what this eta is and this is a stochastic differential equation where f and g are prescribed functions and they are related to a 1 and a 2 as I will write it down in an instant but this eta of t is called a Gaussian white noise a stationary Gaussian white noise and I will explain what that is separately where f of x is essentially a 1 of x and g of x g squared of x is a 2 of x eta of t is a stationary Gaussian white noise I have to say what this means okay eta of t is a random process in time such that all its probability distributions multiple time probability distributions are all Gaussian in shape so that is why it is called a Gaussian noise it is stationary so all its statistical its statistical average and higher moments are all time independent correlation functions or functions only of the time difference etc and it is a white noise in other words it is delta correlated in the following sense equal to 0, 0 mean and it is got a delta correlation delta correlated delta function as an autocorrelation it is clearly the limit the mathematical limit of some physical noise whose correlation time would not be 0 because this implies the correlation time is 0 whereas I would expect for a stationary process if t is bigger than t prime I would expect this correlation to look like this as a function of mod t minus t prime I would expect this correlation to come down in this fashion and this characteristic time scale would be the correlation time of this noise but that is now going to 0 and the amplitude is going to infinity such that in the limit it becomes a data function so it is a mathematical idealization clearly it would have to be justified on physical grounds each time okay for instance in the problem of the collisions of in a gas of particles in a fluid for instance this noise would be caused by all the other molecules colliding against some particular tag particle then the state of t would be the correlation time of that force the random force caused by the collisions all these other guys in the scale on which the particles motion itself is tracked the time scale would be much longer it would remember its memory for much longer than what the noise does so the correlation time of the noise typically would be of the order of nanosecond or a picosecond for instance whereas the correlation time of the velocity of the particle that is being tracked that could be of the order of microseconds so as far as a microsecond is concerned a nanosecond or smaller intervals essentially 0 intervals so in that sense one can justify this approximation okay but each time in any problem when you model this you have to ask whether there is a clear separation of time scales of this kind or not but at the moment from a mathematical point of view the formal point of view this is what this equivalence is so the statement is that a stochastic differential equation of this kind is entirely equivalent to this Fokker-Planck equation for the probability density of this random variable so please take this as a theorem I am not going to prove it here but take this as a theorem we are going to exploit it over and over again now you can see why I call this a drift term because if you did not have this noise at all this is deterministic evolution of this variable under some prescribed function f of x here it may be minus dv over dx we do not care what it is or anything else so this term is indeed describing deterministic dynamics and this is the noise is entirely here in this thing here and that showing up in the second term here okay what is interesting in this problem as opposed to even simpler problems is that this g has x dependence in general so it says given a current value of x of this random variable the way the noise affects it the amplitude of that noise depends on this random variable on this x on the other hand and that is why it shows up here inside here but in the example we looked at in the diffusion problem as a diffusion equation remember this a2 turned out to be a constant so in that case this would have been square root of 2d and that is it so now we can kind of identify what would be the stochastic equation corresponding to delta p over delta t equal to d d2 p over delta x2 this would be equivalent to a stochastic differential equation for x which would be of the form x dot equal to in this case there is no a1 so clearly there is no external force or anything of that kind no drift at all a1 is 0 identically and a2 well this d half d is a2 so a2 is square root of is 2d and therefore g is square root of 2d and that is it this is the stochastic differential equation corresponding to for the position corresponding to the simple diffusion equation in one dimension one can write a formal solution for this guy and that formal solution would be x of t you have to define these integrals minus x of 0 equal to square root of 2d times integral 0 to t dt prime 8 of t prime we can call this x of t not if you like and integrate from t not to t so in that sense it is a plane diffusing particle doing free diffusion the x variable corresponds to the integral of white noise this guy corresponds to the integral of white noise and it is called a Wiener process is it a stationary random process it is Markov because we wrote the mastery equation down I said satisfies a Fokker Planck equation and so on so it is clearly Markov process but is it a stationary Markov by the way it is Gaussian that is something else you have to recognize because we know by hindsight we know the solution of this guy although we did not derive it here explicitly we know the fundamental solution of this is that Gaussian e to the minus x squared over 4 dt which I will come back to talk about so what it is telling us is you are going to hit the particle with a Gaussian white noise that means the distribution of this eta is Gaussian and it is delta correlated stationary it is and Markov then what is the output variable the driven variable x after this integration what properties does it have well it remains Gaussian because its probability density is Gaussian so the shape remains Gaussian that is robust what else happens it is Markov it certainly Markov it is over is this Fokker Planck equation but is it stationary will the stationarity remain does this look like if I if I had x of t not here and this is t not to t does this guy look like a function of t minus t not in general no no certainly not it is not stationary and you already know this because given this diffusion equation what does it imply for this quantity x of t minus x of 0 whole squared what does this become it is the mean squared displacement from some given origin and what is that equal to it is diffusing and therefore what is it equal to 2 dt exactly exactly it is 2 dt it is a function of t it is a function of t so it cannot be stationary because if it is a stationary random process all these moments should be independent of t but here right away tells you it is not stationary immediately we have not computed what the correlation function of x is we have not found what is x of t prime x of t double x of t x of t prime average we have not found that yet but certainly we found what is x of t x squared of t average and that is 2 dt that is just a Gaussian integral so it is not stationary it is not it has stationary increments because you can write this guy obviously as dx equal to square root of 2d 8 of t dt you can write it like that and then of course this is stationary this has stationary increments but it is not a stationary random process by itself or more less rigorously it is derivative is stationary but this function is not the variable itself is not and that you can see directly when you take something which is got stationarity but you integrate it in this fashion integration makes it non-local in some sense so it is not stationary so in general that is a lesson that means you integrate white noise you may not retain the stationarity property but we are going to see that if you put a proper drift you will be able to do this that is what the velocity would do and then it would attain an equilibrium distribution and so on there is another way of saying that this guy is not stationary because it is probability density p of x comma t given an x naught this fellow is decreasing as a function of time and it does not tend to any stationary distribution as t tends to infinity it goes to 0 this Gaussian broadens out over in infinite range the total area under the curve remains 1 but the value at any point is tending to 0 so there is no stationary distribution in this problem the variable is not a stationary random variable either okay. So we will get back to all these things but at the moment I wanted to simply pattern remember the fact that the most general definition of diffusion process could be either this or this does not matter either way now mathematicians do not like to work with these delta correlated noises they rather work with the differential so there is something which smooths it out by integrating so weiner process can be handled more rigorously than this singular object here so you integrate it once to make it smooth and so on but we are not going to pay attention to these niceties we will be careful not to make any mistakes but at the same time we will do this rather heuristically here provided we know there are certain rules which we have to obey and one of them is this that there is this equivalence between the Fokker-Planck equation on the one hand and this on the other and we will see what happens with this now already you can begin to see whether there is going to be a stationary distribution or not by saying if so this thing must have a solution when you make this an ordinary differential with respect to x and if this guy has a solution which is normalizable and so on then you know there is a stationary distribution okay and if you do not have this you do not have this and a is constant then the stationary distribution is to reality itself in this case if at all there is a p stationary it must satisfy d2 p stationary over dx2 equal to 0 but what does that say about p stationary must be a linear function that certainly not normalizable right away you have gone it is finished in an infinite range in an infinite range or even in a semi infinite range it is still not normalizable what would happen if you had a finite range yes then indeed it can be so then you have you do not require integration up to infinity then yes indeed it can it can it is true suppose you are told that p stationary is between these two points nothing more than that I put a diffusing particle here it is like putting a drop of ink inside a beaker of water the ink does not go anywhere but it becomes uniform everywhere intuitively we know this my diffusion so that is what will happen this particles probability density will be uniform will be constant in this boundary provided there is no escape from the ends there is no leakage from the ends so in that case yes indeed because I then I would say p stationary equal to ax plus b and I put boundary conditions at the ends and you discover finally that is 0 and you have just b which is normalized okay so the boundary conditions also play crucial role in the whole thing okay we will talk about these aspects in the next next time we will start with the Fokker-Planck equation and see where we can go