cancelled
International Workshop: Uncertainty and Fluctuations in Thermodynamics
March, 12 -- 13, 2020 — Berlin

Understanding of the fluctuations by which phenomenological evolution equations of thermodynamic origin can be enhanced is the key to a general framework of nonequilibrium statistical mechanics. These fluctuations provide an idealized representation of microscopic details. We consider fluctuation-enhanced equations associated with Markov processes and elaborate the general recipes for evaluating dynamic material properties by statistical mechanics. Path integrals play a central role in the development. Markov processes with continuous trajectories are conveniently characterized by stochastic differential equations and lead to Green-Kubo-type formulas for dynamic material properties. Markov processes with discontinuous jumps include transitions over energy barriers with the rates calculated by Kramers. We describe a unified approach to Markovian fluctuations and demonstrate how the appropriate type of fluctuations (continuous versus discontinuous) is reflected in the mathematical structure of the phenomenological equations. Acknowledgement: This presentation is based on joint work with Alberto Montefusco and Mark Peletier.

I will present work on current fluctuations in stochastic processes with resetting, demonstrating in particular how phase transitions in the large deviations can be understood via a mapping to the classic Poland-Scheraga model for DNA denaturation [R. J. Harris and H. Touchette, J. Phys. A: Math. Theor. 50, 10LT01 (2017)]. This approach can be extended to treat the case where the dynamics between resets depends on the value of a parameter fixed at the last reset and, in this context, I will briefly discuss the application to run-and-tumble models of bacterial dynamics. Finally, I will mention the recent development of a "Thermodynamic Uncertainty Relation" which applies for such run-and-tumble-type processes [M. Shreshtha and R. J. Harris, EPL (Europhysics Letters) 126, 40007 (2019)].

In the first part of my talk I will present the large-time asymptotics of weak solutions to Maxwell–Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. The key elements of the proof are the existence of a unique detailed-balanced equilibrium and the derivation of an inequality relating the entropy and the entropy production. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. In the second part of my talk, I will present some recent results on uncertainty quantification for the underlying multi-species Boltzmann system. This talk is based on a joint work with Bao Q. Tang and Ansgar Jüngel, as well as on a joint work with Liu Liu and Shi Jin.

We discuss dynamical large deviations of quantum trajectories in Markovian open quantum systems, as described by Lindblad evolution equations [1]. In particular, we discuss two possible ways that these systems can be unravelled, in order to obtain a stochastic evolution equation for a pure quantum state. Averaging this evolution gives the Lindblad equation, but the full stochastic process carries additional information about the statistics of measurements in the environment of the quantum system. We discuss large deviations of time-averaged quantities in the unravelled process, including joint fluctuations of the empirical measure and empirical current (which is called level 2.5). We derive a thermodynamic uncertainty relation for a particular class of these systems, which are quantum reset processes. We discuss possible consequences for the interpretation of Lindblad equation as gradient flows [2].
[1] F Carollo, RL Jack and JP Garrahan, Phys Rev Lett 122, 130605 (2019)
[2] M Mittnenzweig and A Mielke, J Stat Phys 167, 205 (2017)

In this talk we study a system of (hard-core) interacting Brownian needles. Unlike point particles, the finite size and shape of each needles has an influence on the evolution of the system. We explore the effects of excluded volume and anisotropy at the population level for these systems. Since needles exclude less volume if aligned, can excluded-volume effects alone induce order in the system? Starting from the stochastic particle system, we derive an integro-differential equation for the population density using the method of matched asymptotic expansions and conformal mapping. We present numerical simulations of the particle- and population-level models, and discuss the limit of large rotational diffusion. I will also discuss, more generally, the connection between the entropy of the particle system and the gradient-flow structure of the population-level PDEs.

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method for taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities. A generalization of the method will be shown. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution.

Many stochastic particle systems have well-defined continuum limits: as the number of particles tends to infinity, the density of particles converges to a deterministic limit that satisfies a partial differential equation. In this talk I will discuss one example of this. The particle system consists of particles that have finite size: in two and three dimensions they are spheres, in one dimension rods. The particles can not overlap each other, leading to a strong interaction with neighbouring particles. Such systems of particles have been much studied, but for the continuum limit in dimensions two and up there is currently no rigorous result. There are conjectures about the form of the limit equation, often in the form of Wasserstein gradient flows, but to date there are no proofs. We also can not give a proof of convergence in higher dimensions, but in the one-dimensional situation we can give a complete picture, including both the convergence and the gradient-flow structure that derives from the large-deviation behaviour of the particles. This gradient-flow structure shows clearly the role of the free energy and the Wasserstein-metric dissipation, and how they derive from the underlying stochastic particle system. The proof is based on a special mapping of the particle system to a system of independent particles, that is unique to the one-dimensional setup. This mapping is an isometry for the Wasserstein metric, leading to a beautiful connection between limit equations for interacting and non-interacting particle systems.
This is joint work with Nir Gavish and Pierre Nyquist.

The dynamical variables φ of a classical system, undergoing stochastic stirring forces, satisfy equations of motion with noise terms. Hence, these φ show a stochastic evolution themselves. The probability of each possible realization of φ within a given time interval, arises from the interplay between the deterministic parts of dynamics and the statistics of noise terms. In this work, we discuss the construction of the stochastic Lagrangian out of the dynamical equations, that is a tool to calculate the realization probabilities of the variables φ as path integrals. In this formulation, the study of classical statistical dynamics can benefit from all the techniques developed in Quantum Mechanics of path integrals; moreover, as the path integral is expressed in terms of a Lagrangian, the invariance properties of the system become transparent. After a coincise review of the stochastic Lagrangian formalism, some applications of it to physically relevant cases are illustrated. Then, the advantages and maturity of this approach, and its expected future developments, are outlined.

In this talk I will use the concept of entropy power to derive a new one-parameter class of information-theoretic uncertainty relations for pairs of observables in an infinite dimensional Hilbert space. This class constitute an infinite tower of higher-order cumulant uncertainty relations, which allows in principle to reconstruct the underlying distribution in a process that is analogous to quantum state tomography. I will illustrate the power of the new class by studying Schroedinger cat states and the Cauchy-type heavy-tailed wave function. Finally, I will try to cast some fresh light on the black hole information paradox.
[1] P. Jizba, J.A. Dunningham, J. Joo, Ann. Phys. 355, 87 (2015).
[2] P. Jizba, J.A. Dunningham, A. Hayes, Y. Ma, Phys. Rev. E 93, 060104(R) (2016).

Stochastic transport systems are known to have an interplay between the current, current fluctuations and their entropy production rate. In this talk we will cover two results. I) When can time-periodically driven diffusive systems outperform nonequilibrium steady state bulk driven systems. II) How to define a transport uncertainty relation for mesoscopic (athermal) systems and what are the resulting applicative implications.

The properties of the liquids in the vicinity of an interface might be very different from those measured in a large volume. Near the interfaces, the nature of the interactions can dramatically affect both dynamic and static liquid properties. Significant deviations from homogeneous temperatures have been experimentally highlighted revealing that similar to external fields such as electric, magnetic, or flow fields, the vicinity of a solid surface can preclude the liquid molecules from relaxing to equilibrium, generating located non-uniform temperatures away from the solid surface [1]. This effect reveals also a high degree of connectivity between liquid molecules in agreement with the identification of elastic correlations (low frequency shear elasticity) in various liquids and viscoelastic fluids [2].
The high degree of connectivity between liquid molecules (elastic correlations) challenges the molecular approaches and imposes the consideration of collective modes similarly as in solids. More parameters (lengthscale dependence, interfacial force boundaries, compressibility effects) have to be taken into account making the liquid characterization more complex but also richer of potential applications. The effects are particularly revealed at the sub-millimeter scale, impacting microfluidics in particular the flow mechanisms of physiological fluids.
Keywords: liquids, viscoelastic fluids, solid wall, wetting, long range interactions, collective effects.
References:
[1] L. Noirez, P. Baroni, JF Bardeau, Highlighting non-uniform temperatures close to liquid/solid interfaces. Appl. Phys. Lett. 110, 213904 (2017) DOI:10.1063/1.4983489
[2] L. Noirez, P. Baroni, H. Mendil-Jakani. The missing parameter in rheology: hidden solid-like correlations in liquid polymers and glass formers. Polym Int (2009) 58:962

Closed and homogeneous in the reduced statistical operator exact and approximate time-convolution and time-convolutionless differential master equations were derived by means of the projection operator method for the reduced statistical operator of a multi-level quantum system with finite number N of quantum energy eigenstates interacting with arbitrary external deterministic fields and dissipative environment simultaneously. Unlike conventional master equations derived by the projection operator method for the quantum systems of this kind so far, these equations are homogeneous in the reduced statistical operator while, at the same time, taking into account initial thermodynamically equilibrium correlations between the multi-level system and the environment. Consistent representation of the multi-level system evolution in terms of the SU(N) operator algebra allows for efficient numerical analysis of these and conventional master equations derived for various physically realistic quantum models.
This work is supported by the Program of the Presidium of the Russian Academy of Sciences n.01 "Fundamental Mathematics and its Applications" under grant PRAS-18-01.