 Non-Makovian systems a lot of none Thank you everybody, thank you to organize this nice conference. Yeah, that's the topic of my talks Yeah, I will start to say some motivation why we are talking study The mean first pastime. So as a first things quite a lot of natural system can be a then rescale a study in One deep potential in particular as a barrier crossing in particular some rare events as chemical reaction and protein folding and another Motivation is that's a normal the system arm are not that the equilibrium and in particular from that's article from boost a month where they are study some folding of proteins they So some delay in that's folding and in particular that's delay time is exponential so here from the dots the data and The curve is some exponential fit and from the dash line is the line that they expect if the system is fine That's the equilibrium. So I will introduce my system as we already Here a little bit during today I will use a general I Lunge of an equation where we know that the random force is correlated with the kernel and in particular I will use Decay exponential kernel where gamma is the friction and tau is the memory time of our system and My potential as I say in the introduction it will be a double well Symmetric potential, but as I say in my title we are interested to study System that are out of the equilibrium and to do that one we violates the fluctuations dissipation here And for that reason we use two different functions tau V on gamma V on gamma R They function have the same Yeah, the function They function have the same Shape but two different memory time and When we are out of the equilibrium tau V is different or why we change only the tau and we don't scale for example the friction Coefficient because we know that every time that we rescale the friction coefficient We can always redefine easily effective temperature that's taken count of that Difference and bring back our system at the equilibrium. So starting from that Generalize Langevin equation we Fourier transform the autocorrelation function and here we introduce approximation. So we will not consider our double well Potential my simple Quadratic potential so we Fourier transform we calculate the autocorrelation function and We observe that the two memory time that we are interesting tau V and are In function of gamma V and gamma R that are in our Final expression in fact we put all the terms that we know inside and we arrive at that expression That's the special it looks a little bit ugly But we can rescale and find some effective parameters so we go to rescale and we We call the new parameter as a new effective friction new mass Effective mass a new Effective temperature that inside the beta F That's the effective parameter that we We investigate and it's nice at the beginning to see how that's effective parameter are changed when we are in a new Markovian system because respect at our normal Oscillating dump our oscillating model that's effective parameter are already Changed where we introduce the memory, but our effective temperature is changed only where we are out of the equilibrium because it's depend on the the two memory time to R and to V that's C1 is our only our constant that we will fit with our simulation to take account That's the model So the analytical calculation that we are doing we are doing for a quadratic potential But in our simulation, we will use double wall potential and in total we will have three constant parameter Yeah, the equilibria is already say beta effective is equal the the temperature effective is equal our temperate of the system at the beginning so Mimi's Mimi means I the denominator of Of the autocorrelation function we studied the two limits of the gamma effective so we have a expression for low friction limits and Expression for high friction limit as I say before also in that two case we add some constant that we will fit so now that we know the our Effective parameter we can come back A Markovian system and we can find to So our goal is to looking for a special or me first for such time but no Markovian and non-equilibrium and to do that we start from the much covina system Well, we know that the the memory current the memory time is equal to zero and we know the two exact expression from Kramers and for high friction and Low friction in the plot we see in with the red curve the two limits that Dershow with the law the yellow line is Mexico Formula that's a little bit complicated formula. I would like to see something More easy and with the star our simulation the dash at the black line is the is a formula that we obtain were Simply we sum the the two limit and we add the Intermediate limit the a cross cover Markovian term That's our data to memory time that I use a mess system That's our day in Asia and diffusion time that they used to rescale my time So starting from the Markovian system we can go to Check the no Markovian system where we introduce a Instead of the normal mass and friction our new effective mass a Effective friction and we obtain this formula here still we are at the equilibrium So to V is equal to R. So we have only one memory time for that the beta Coefficient it doesn't change the temperature doesn't change and we also have that formula fit Fit very good with our data in particular that's is the expression when we consider the term and my effective of a gamma effective and And then the effective Friction at a high friction limits we can consider as gamma because fitting the third Parameter so C3 we observe that has a very small value So I would like to point another a couple of things Yes, the first things is that's equation that we found out from Analytical collation is have as exactly the same shape of equation that was found in a more heuristic way from Julian coupler in XPST in our group and Other part is that this term the negative term That's the last term This come from omega Mass effective of a gamma effective It's a negative term that I would like to don't have so to avoid any problem to obtain Negative time and we sum with the high friction Term for that we have this expression one over and that's term is the responsible of the speed up regime so the regime where the memory brings the memory Brings the memory the mean first past the time be be smaller respect the Markovian limit Okay, that's all the parameter that I use for for that slide so now that we found our Expression for the no Markovian system we will go to consider the no Markovian system as non equilibrium and Also in this case we consider the two limits when we are in a low friction and the high friction limits And we observe that at the end we found that in Some limits that are True in our system. That's the two expression of more or less the same behavior and for that We will all introduce one new Effective temperature that that's better than an equilibrium that is equal at our over to be square to check if that's parameter exactly It's a good parameter for our system we will consider the position distribution The the position velocity distribution in that spot. I show the position distribution So in a there is the position distribution from our simulation Where in B? I rescaled for that's effective parameter and in C and D. I plot that's effective parameter respect our respect Toby and we observe that the in the star with the star which we plot exactly the Rescaded parameter and all the line that Describe that show this three different functions are perfect agreements With the data and also perfect remit with them and we do the same things for this velocity distribution And we found the same as we expected the same result so at the end we insert that's better non-equilibrium inside our formula and We observe as the line that show the formula qualitative qualitatively describe our simulation and the nice things to observe is in this case We are not only to modify the constant after the Arrhenius term We modify the Arrhenius term and in particular they have exponential sorry The exponential behavior so when we increase our we observe a very large increase of the mean first passage time Then the same result that I show in the motivation motivation at the beginning so in conclusion We found out some effective parameter friction mass and temperature Solving the non-equilibrium non-arm non-marcovian harmonic oscillator We found out a formula for the mean first pass time where the non-equilibrium effect strong strongly the The time because it's changing the exponential part and for that's the Arrhenius constant and in conclusion We observe that that's effective temperature describe the position at the velocity for a system For a barrier cross-system stem re also really far from the equilibrium. Thanks Thank you very much for the talk Are there any questions in the audience? So I have a question about the So generally with these with these problems when you study these mean first passage times Often they turn out to be quite untypical for the within the distribution So my question is can you say something about higher moments or about how the distribution? Changes when you increase memory in the system Yeah, in particular. Yes, because if you watch that so actually that says We immediately see how the memory and non-equilibrium going to change also the distribution but yeah We can see but Qualitatively does it spread or does it start? so so So we observe from the distribution that I show how they change but always we can rescale it So at the end of we know how they change the Distribution Yeah, I think also we can find out I think how it's so I didn't check Precisely, but I think it's possible to observe. I think that is the second part or next step That's also we have to investigate After that Yeah, so you showed that there is a regime where memory can speed up the barrier crossing, right? But that wasn't the equilibrium case if I'm yeah, okay You go to the non-equilibrium case. I assume that you can like enhance this effect. Have you looked at this? Were you interested in that? so not because to be honest we We thought so that's the most interesting party was that really we changed that arenas fact and Yeah, yeah, that's Because we more or less we observe more how the non-equilibrium brings a higher Level, but sure probably here we see because if we observe the scale Probably when we are a smaller scale, we observe also the speed up. Yeah Yeah, I was wondering if you could find the regime where you have like a very No, I don't I don't think so to be honest. Okay. Okay. Thank you because if we assert also here It's quite interesting, but it's quite as more regime and yeah, probably but I don't think that's it's and then it's really Important as a factor respects the fact that we see has they really Modified I don't know. That's my impression, but I don't know There's a question online So I'm just gonna read it out in principle You could map your system to my curving one in an extended state space. What would I predict for the mean first passage time? Yeah, so injera to be honest, I don't know but Yeah, I always remap with some mapping but predicts So we already know which is the normally that's the point that we compare how that's the mean first time They know Markovian and the non-equilibrium change in respect of the Markovian time. So that's I think it's really a Yeah, we can predict how Quite change the the parameter that we are in our in the Markovian. I think that's more or less the deal But I don't know Did this address the question? I don't think so Not me. Okay in that case. Thank you very much