 Iyer Kolkata as a PhD student from Orin Banerjee's group and first I'd like to thank all the organizers for having me here. So I will just talk about energetic fluctuations of the fluctuations of energetics of a trap brownian particle in active like viscoelastic bath. So by now we all know that any microscopic processes in microscopic regime like the position fluctuation of brownian particle in time dependent trap or any kind of stochastic pump or even the walking of kindness in motor are very driven by thermal fluctuations and stuff like that and that's why every time if you realize the system you will get different different trajectory. Then if you do any trajectory dependent thermodynamics and try to quantify work and entropy and it like that every quantity will have a distribution and rather than a single value this will have a distribution and there are this fluctuation theorem which tells us about the probability of the positive fluctuations and the probability of the negative fluctuations and they say that the probability of the positive fluctuations are exponentially more. So and if you take the log of that you will get a basically a straight line curve depending on the curve and on this line there is this work fluctuation theorem which is like relates the positive work fluctuation to the negative work fluctuation and this fluctuation theorem is very well studied in many different scenarios starting starting from experiment in colloidal substance where a particle is trapped in optical twigel and moved by some kind of external random force and so that depending on the strength of the noise this relation is validated and after if the strength of the noise is very high this noise this this relation is violated. Then there are other theoretical results who talk about this the violation and where this thing happened and rather than the colloidal system many other situations like energy flux in wave turbulence or any electrical register think there also these kind of relations are studied and the violations are well studied also. Now we wanted to look into the work fluctuation as well as the fluctuation of any energetics in a system where the bath is fiscoelastic and why fiscoelastic because so the fiscoelastic material is like this bath is not only dissipated energy but it can some sense it can store energy that leads to the large relaxation time of the fluid and in recent times there are this interesting observation like this enhancement of tremor state transition and some theoretical estimates of that if the bath is considered like fiscoelastic theoretically the efficiency of a Carnot engine can or say efficiency of a heat engine can leads to a Carnot efficiency rather than there are many other thing if you have a fiscoelastic bath and you have active particles they do so circular motion and their rotational diffusion and increases more importantly there are several papers that shows that in different biological processes are felicitated due to fiscoelastic that but still now there are some questions that needs to be answered that like how this fiscoelasticity affecting the non-equilibrium condition or how it can optimize the output of such microscopic processes and even you can consider like inside the cell the medium is fiscoelastic and if you put a probe particle there that will feel the active fluctuation due to the active background inside the cell which you can abstract like a microscopic particle in a harmonic trap and which trap position is fluctuated to understand this type of process we realize experiment that using optical vision where we trap a microscopic particle in in say polyethylene oxide which is of his fiscoelastic material and which relaxation time is larger than any viscous material or something like so relaxation time cell here we use the material which relaxation time scale is like 0.018 a second and to mimic the activity in the background what we did we moved the mean position of the trap using a exponentially correlated noise through this acoustic modulator so that you can see as we increase the strength of the external noise the system goes to outer equilibrium and you can see this steady state current in the background okay and to understand this system analytically we write down this the generalized langevin equation with a memory kernel and the mean position of the trap is moved like a on-stain rule and back now okay and this memory kernel which has a delta correlated term plus a exponential correlated term this tau which related to the relaxation time scale of the fluid okay so due to this kind of thing this langevin equation is non-marcovia and to understand or to solve this thing analytically to understand the dynamics of this analytically what we do we introduce a auxiliary variable that so that this non-marcovian system can be written down in a higher order marcovian system so that we can get the steady state probability distribution of the particle in terms of mean and variance of the structures so first we look into the variance of the particle what we first notice that the variance of the particle changes linearly with the strength of the noise which is here A and also we look into that in from the mean square displacement and as we see that mean square displacement is increases for higher noise strength as expected it is increases on the higher noise strength then what we what we notice that for a certain fixed noise strength which is determined by this parameter theta that we saw that for a viscous case for a viscoelastic case the variance or you can say the effective temperature of the particle is increased than the viscous bar and this nature is here also in the in the steady state probability distribution too also we notice that this variance of the position fluctuation of the particle and does not change monotonically with the relaxation time of the fluid but it changes non-monotonic fashion and after some point it does saturate does not mean it if you increase the relaxation fluid the sigma the variance does not keep on increasing but it's saturated at some point next we look into the energetic first we look into the work done on the trapped particle and which is defined like that this is just a force times the displacement and force times velocity over dt so this is this and then we able to calculate the mean of the work done and we find out that the rate of mean work done is more in case of a viscoelastic bath than the viscous bath okay and here also with relaxation timescale of the fluid we get similar nature that we saw in case of a in case of a viscoelastic okay so then what we saw that the entropy production rate of the system is found to be exactly equal to the work done on mean work done on the work done of the trapped particle that's because if you look into the entropy production current there is only the time extensive part is the input work done that is the only extensive part so the mean of this thing is equal and we kind of expected that the mean rate of entropy production or as well as the work done is increases in case of viscoelastic bath because in viscoelastic bath there are the increase of interaction of the particle with the bath due to a different kind of polymer present in the viscoelastic bath and as we know from recent literature that we know the interaction increases the entropy production rate so that's why in viscoelastic bath entropy increase or the work done on the trapped particle is increased then we look into the fluctuations so to understand the fluctuation of the thing first we look at the single step work distribution so here we to theoretically understand that what we do we take care of the fact that this x0 which is the mean position of the particle and x dot are two Gaussian random variable which are cross correlated and we so then the single step work fluctuation can be explained can be expressed in terms of a skewed this 0th order basal function modified basal function where this beta w s and alpha w s term can be expressed in terms of the mean and variance of the particle and then what of the work so we saw that our theoretical prediction is matched well with the experimental data from where we calculated single time work done and also if we look at the work fluctuation theorem it also matches with the theoretical points then we look into the cumulative work distribution but for the cumulative thing the work series is already correlated right so we kind of approximated the correlations of that approximated the correlation in the mean and variance of the particle and we kind of approximated this this the fluctuation in terms of the modified basal function and such that it matches well with our experimental experimental experimentally measured input work done and what we notice that at long time the cumulative work done the in the viscoelastic cases for a certain strength of the knowledge the negative work done probability of getting negative work done is diminished compared to the viscous case okay I must say that to get this probability distribution of this work done there are more general method like the last deviation of probability or the path probability measure where also you can get this type of long time behavior in a more general way but we we just followed this to get the approximated result and it fits well with our effort with our experimental data and we also looked into the fluctuation theorem here and what we notice is there is that that the fluctuation theorem for a viscoelastic case is violated with a lower noise strength than the viscous case okay so for if we notice that this even the in a viscous case even even the fluctuation theorem is violated it's valid up to let's say theta equals to 0.62 with our data but if we notice it's actually violated for the viscoelastic for the very lower noise so as a whole what we notice that we saw the viscoelasticity of a medium increases the input work done on the trap particle and that way the negative work fluctuations are also diminished in case of a viscoelastic bump and and also there are the strong violations of the work fluctuation theorem in viscoelastic so I would like to thank the group and I'd like to thank my collaborator Professor Supriya Krishnamurthy and Srikant from Stanford so I would like to thank the my funding and thank you for this any question thank you very much do we have any quick questions I I have a question so if you talked about the entropy production going up in the viscoelastic case yes yes my my sort of intuition for when you get entropy production from a system that you're jiggling right effectively you're you're jiggling the system from the outside yes entry production is low if the response is really slow compared to your jiggling because the thing just doesn't move and then you don't get an entry production is low if the thing responds really quickly because then it's like you're doing quasi-static jiggling but we get high entropy production when the system is responding to your perturbation but not keeping up with it and it feels like the the having a long memory kernel or whatever it is is basically making that region where it responds but not quickly much bigger does that match with your intuition of why the thing is generating more entropy yes kind of but we think that so there is this thing so if I go back to that slide so if you notice so in a viscoelastic bar there are a different kind of polymers and stuff like that which increases the interaction of the the suppose the graph particle with the bar so this kind of interaction actually increases the entropy production date so increase in top interaction can increase the entropy production date so we are thinking by that way but I think your intuition is also correct so because of this memory kernel the interaction is actually increased okay thank you very much for your talk let us and it's nice to see a combination of theory and experiment there