 session and as you know this session we will have three speakers and each talk will be 30 minutes and after these three talks we will have also a discussion time for 30 minutes so you can keep to ask your question or discuss your question in this next discussion ground and after that we will have 15 minute breaks then the second session will start okay the first speaker it was from University of Tokyo hi Soseko you can share your thoughts okay so can I could you see my screen is it okay could you see my pointer yes yes yeah okay so so thank organizers for inviting me and organizing a very exciting workshop I'm Sosuke Ito from the University of Tokyo Japan and today I would like to explain a new decomposition of the entropy production rate okay so let's start so this is a list of the collaborators Andrea, Shinichi, Kohei, Artemis and Muka and several papers in these topics have been published okay so at first I would like to introduce conventionally the composition of the entropy production rate this is well known as the Hata no Sasa decomposition or adiabatic no adiabatic decomposition so entropy production rate is decomposed into the two non-negative parts excess entropy production rate and housekeeping entropy production rate and in a no equipment steady state excess entropy production rate is zero and housekeeping entropy production rate is given by the entropy production rate in a steady state and this quantity is characterized by the cycle flow in a shiitake back network theory so this is the example of the Hata no Sasa decomposition for the Fokapran equations so we consider the non-equivalent steady states and excess entropy production rates and housekeeping entropy production rate are defined by using the steady state probability distributions so yeah it could be okay for the Fokapran equations because we can assume the existence of the unique and asymptotic stable non-equivalent steady state but it could be a problem in chemical thermodynamics dynamics is given by the non-linear rate equation in chemical thermodynamics because we have a multiple non-equivalent steady states and these non-equivalent steady states can be unstable so in a chemical thermodynamics such kind of the decomposition is only defined for a very special case where the asymptotic stable non-equivalent steady state exists okay so now I would like to propose another decomposition of the entropy production rate namely the geometric composition without considering a non-equivalent steady state so in our decomposition excess entropy production rate means dissipation by the same type of evolution driven by a potential and housekeeping entropy production rate means dissipation by a cycle flow which does not affect the time evolution so this idea of the decomposition is originated from the geometry in optimal transport theory so I would like to briefly explain the optimal transport theory and this is a very old-story in mathematics and optimal transport problem is the following so we would like to transport a probability distribution from p to q by using the map t and the transport cost for each point depends on the distance between x and y for example and what is the map t to minimize the all the cost if we consider this optimization problem we can define the distance in the probability space that is a washerstein distance so this is a solution of the optimization problem and this quantity is very interesting in some dynamics because this quantity is strongly related to the free energy for the Fokker-Planck equation okay so next I would like to explain the optimal transport based on the dynamics continuity equations so we consider the following continuity equations and at washerstein distance is given by this optimization problem and with a fixed initial and final distributions and this optimization problem can be solved and the solution is given by this velocity fields velocity fields is given by the gradient of the potential and we can yeah obtain this equation that is related to the pressureless Euler equation in fluid mechanics without velocity so intuitively speaking optimal transport is a kind of the transport without velocity okay now we'd like to consider the binomial formula for the Fokker-Planck equation so if we consider the definition of the entropy production rates then we can obtain the lower band on entropy production that is given by the washerstein distance so nowadays this band is known as the kind of the sum of the speed limit and we have revisited this expression from the viewpoint of the geometry if we consider the infinitesimal time evolution then we can obtain the lower band on an entropy production rate so and this equality was when yeah velocity fields is given by the gradient of the potential and we now identify these washerstein parts with excess entropy production rates and this difference with housekeeping entropy production rate okay so this is a key figure in this talk so we now consider the Fokker-Planck dynamics and the dynamics is driven by the non-conservative force so then it is not optimal but now we consider the optimal transport driven by the gradient of the potential and which gives a same time evolution and we consider the difference and the difference is given by the cycle flow because by by using two inequality we can obtain this equation so then yeah we can consider the dissipation for optimal transport and dissipation for this both this is cycle flows okay and this decomposition is strongly related to the geometry so if we introduce a inner product then this decomposition is a kind of the pitabary and theorem for this right triangle and this equation was because of this orthogonality and I would like to remark that the mathematical is the same the composition has been proposed by mass and netoteny based on the this equation and this equation corresponds to this equation and this orthogonality is originated from the orthogonality of the two spaces kernel of the minus type versions and the image of the gradient so this quantity is in the image of the gradient and this quantity is in the kernel of the minus type versions because we can obtain this equation right and by using this orthogonality we also can obtain the orthogonal decomposition for the Karibach-Ribera divergence that is a orthogonality in information geometry so now we consider the path probability distribution for the modified dynamics the meaning of the modified dynamics is given by the velocity field but new prime and if we consider the Karibach-Ribera divergence for two path probabilities and then this orthogonality provides this equation this is a kind of the generalization of the pitabary and theorem in the information geometry and this equation gives the same decomposition of the entropy production rate okay this is the story for the Fokker-Planck equation next I would like to consider the generalization for the rate equation in chemical sum dynamics uh in chemical sum dynamics we consider the following rate equation this is a generalization of the continuity equations because this matrix S stoichiometric matrix is a kind of the differential operator so which corresponds to the minus divergence minus divergence and I would like to remark that yeah we this rate equation is a kind of the generalization of the master equation because yeah if we consider the instance matrix as a stoichiometric matrix then mathematically we can obtain the master equation okay anyway so now we consider the entropy production rate in a chemical sum dynamics yeah define this and now we consider the orthogonality of the two spaces kernel of S and the image of the S transpose and in this S corresponds to the minus divergence and S transpose corresponds to the gradient and now we consider two generalizations of the geometric decomposition for the rate equations first we consider the geometry of the inner product and next I would like to consider the geometry of the carbacrylate divergence okay this is the first geometry so now we consider the inner product given by the onsite matrix and the entropy production rate is given by the square of the norm norm f and we consider the optimal transport for the rate equations that means that this j star gives the same time evolution and j star is given by the kind of the gradient potential right if we consider this optimal transport based on this orthogonality we can obtain the geometric decomposition right because this orthogonality corresponds to this orthogonality so then we can obtain this Pythagorean theorem and we yeah define the excess entropy production rate and the housekeeping entropy production rate for the rate equation I think that this definition of the entropy production rate excess entropy production rate is good compared to the Hatano-Sasa's excess entropy production rate for example if we consider the brass letter we only have one unstable non-equivalent steady states so if we consider the Hatano-Sasa's excess entropy production rate this quantity can be negative but if we consider our geometric decomposition and if we consider the excess entropy production in our way then this quantity cannot be negative and yeah this excess entropy excess entropy production rate reflects the behavior of the dynamics okay finally I would like to and yeah explain the another decomposition information geometric decomposition so yeah if we consider chemical rate equation we cannot define the pass probability distributions but now we consider the unidirectional flow and consider the generalized carbacryber divergence in this way so then we can prove that entropy production rate is given by the generalized carbacryber divergence for one way flux by way of flow and so then we can also consider the optimal transport problem for this carbacryber divergence now this is a we consider this rate equation as we yeah redefined we define the stoichiometric matrix for the one way or unidirectional flow and then we now consider the this flow there which is given by the kind of the gradient of the potential right so if we consider the this also monarities also monarity of the two spaces kernel of the s-childer and the image of s-childer transpose then we can obtain the this generalization of the Pythagorean theorem so then yeah the entropy production rate is decomposed into two non-negative paths and this quantity can be a housekeeping entropy production rate and I would like to remark that yeah the two generalizations of the excess and housekeeping entropy production rate are not equivalent but numerically we can see that there is some inequality between them and the information geometric excess entropy production rate is always less than the on saga projective excess entropy production but it's just a numerical yeah evidence so we cannot prove it okay so finally I would like to uh uh say the application of the geometric composition to the study of TURS yeah we believe that such kind of the decomposition it might be useful to study lower bound on the entropy production rate so examples for the four-cup rank equations this excess entropy production rate gives a good upper band on the speed of the observable and if we consider the rate equations we also can obtain the good upper tighter upper band on the this speed and we also can ensure that if we consider the information geometric entropy production rates uh excess entropy production rate which provides a TUR for highly irreversible process that is uh inequality and this is a TUR for the highly irreversible process okay that's all so in our talk in my talk uh I proposed we proposed a geometric decomposition of the entropy production rate and in our decomposition uh the excess entropy production rate means dissipation by the same time revolution driven by a potential and housekeeping entropy production rate means that dissipation by a cycle flow which is not affected time revolution and we generalize the geometric decomposition for non-linear related equation where unique uh unstable non-equivalent theory says does not exist generally and geometric decomposition may be useful at least to derive the TURS thank you thank you sasuke for this nice talk and uh yeah there are labs also so are there any question I think I saw some question in chat yeah maybe we start from chat there's a can can you see the question in chat that's now I do not know the steepest entropy assent formalism uh but uh our decomposition is based on the sorry um my sorry I here shows this right okay ah sorry can I could you see my yes yes we see your presentation yeah so yeah my decomposition is based on the um minimization of the entropy production rates so then yeah we can obtain a kind of the variational formula and in this um we obtained this triangle right triangle so then we can obtain a uh kind of the variational formula for excess under housekeeping entropy production rates so we can obtain the excess under housekeeping entropy production rates by using the projection theorem in geometry um so is it okay for you so is it yeah yeah Rohit do you want to add anything to your question because I see your hand yeah yeah thank you thank you for the answer yeah that will do for the moment I have another question though it is that in the excess entropy production rate you said that it can be sometimes negative so how do you explain that given that it is a ratio of a square and some inner product so you mean that uh negative means that this result yeah yeah the Hatano-Sasa excess yeah yeah for example uh the Hatano-Sasa excess entropy production rate is given by the uh uh the Carbacriber divergence between the non-equivalent steady state and the current probability distribution or current states and the excess entropy production rate is given by the time derivative of this quantity so then for example the distance between the yeah non-equivalent steady states and this current value can be changed and it can be positive and it can be negative that is why yeah Hatano-Sasa excess entropy production rate is not so useful in this system but in our form formulations uh we do not uh consider the this non-equivalent steady states so then yeah it could be always positive okay thank you very much yeah okay there's another question in the chat John from John Sautamba are there any specific considerations or challenges when applying geometric decomposition to non-linear or non-equilibrium systems uh so for example this is a kind of the non-linear program and yeah we also consider the another uh applications that are now uh yeah I'm preparing um but anyways uh in my understanding um we can obtain the good decomposition of the entropy production rates in terms of the geometry and this yeah strategy might be useful for the several systems for example the non-linear program yeah but anyway this is a kind of the non-linear program so okay thank you are there any other question check there is one more I think and on I see you Lou on this page how would other definitions of excess heat behave on this page how would uh so what do you mean by other definitions uh so for example uh you mean that's uh information geometric uh excess entropy production rates for anyway so yeah if we consider the yeah excess information geometric excess entropy production rates uh for this raster systems the behavior is very similar uh okay I see like defined by mona and panic point okay I got this uh so now uh I do not have any answer now but uh in my understanding uh it might be difficult to generalize such kind of the excess entropy production rates uh for the non-linear uh dynamics uh um because uh we do not assume the existence of the uh unique and uh asymptotically stable yeah no equipment still is that's now I don't have any actually okay thank you okay I think we can continue also questions in discussion time so now um we can go to next speaker I think