From the Horowitz-Esposito stochastic thermodynamical description of information flows in dynamical systems [J. M. Horowitz and M. Esposito, Phys. Rev. X4, 031015 (2014)], it is known that while the second law of thermodynamics is satisfied by a joint system, the entropic balance for the subsystems is adjusted by a term related to the mutual information exchange rate between the two subsystems. In this article, we present a quantitative discussion of the conceptual link between the Horowitz-Esposito analysis and the Liang-Kleeman work on information transfer between dynamical system components [X. S. Liang and R. Kleeman, Phys. Rev. Lett. 95, 244101 (2005)]. In particular, the entropic balance arguments employed in the two approaches are compared. Notwithstanding all differences between the two formalisms, our work strengthens the Liang-Kleeman heuristic balance reasoning by showing its formal analogy with the recent Horowitz-Esposito thermodynamic balance arguments.
The Kullback-Leibler inequality is a way of comparing any two density matrices. A technique to set up the density matrix for a physical system is to use the maximum entropy principle, given the entropy as a functional of the density matrix, subject to known constraints. In conjunction with the master equation for the density matrix, these two ingredients allow us to formulate the second law of thermodynamics in its widest possible setting. Thus problems arising in both quantum statistical mechanics and quantum information can be handled. Aspects of thermodynamic concepts such as the Carnot cyc…
In this article, we present an approach for the thermodynamics of phase oscillators induced by an internal multiplicative noise. We analytically derive the free energy, entropy, internal energy, and specific heat. In this framework, the formulation of the first law of thermodynamics requires the definition of a synchronization field acting on the phase oscillators. By introducing the synchronization field, we have consistently obtained the susceptibility and analyzed its behavior. This allows us to characterize distinct phases in the system, which we have denoted as synchronized and parasynchr…
We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a the…
The transfer entropy is a well-established measure of information flow, which quantifies directed influence between two stochastic time series and has been shown to be useful in a variety fields of science. Here we introduce the transfer entropy of the backward time series called the backward transfer entropy, and show that the backward transfer entropy quantifies how far it is from dynamics to a hidden Markov model. Furthermore, we discuss physical interpretations of the backward transfer entropy in completely different settings of thermodynamics for information processing and the gambling wi…
Thermodynamics is continuously spreading in the engineering practice, which is especially true for non-equilibrium models in continuum problems. Although there are concepts and approaches beyond the classical knowledge, which are known for decades, their mathematical properties and consequences of the generalizations are less-known and are still of high interest in current researches. Therefore, we found it essential to collect the most important and still open mathematical questions related to different continuum thermodynamic approaches. First, we start with the example of Classical Irrevers…
In this paper, we use quadratic forms diagonalization methods applied to the function thermodynamic energy to analyze the stability of physical systems. Taylor's expansion was useful to write a quadratic expression for the energy function. We consider the same methodology to expanding the thermodynamic entropy and investigate the signs of the second-order derivatives of the entropy as well as previously to the thermodynamic energy function. The signs of the second-order derivatives to the Helmholtz, enthalpy and Gibbs functions are also analysed. We show the immediate consequences on the stabi…
Developing the non-equilibrium thermodynamics of friction is required for systematic design of low friction surfaces for a broad range of technological applications. Intuitively, the thermodynamic work done by a material sliding along a surface is expected to be partially dissipated as heat and partially transformed into the change of the internal energy of the system. However, general non-equilibrium thermodynamic principles governing this separation are presently unknown. We develop a theoretical framework based on the transition state theory combined with the conventional Prandtl-Tomlinson …
We present a study of phase transitions of the Curie--Weiss Potts model at (inverse) temperature $β$, in presence of an external field $h$. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the $β$--$h$ plane as well as the magnitude of the jump of the magnetization (for $q \geqslant 3)$. The signature of the latter aspect is characterized here by the presence or not of a giant component in the clusters of a Fortuin--Kasteleyn type representati…
Aspects of the modern dynamical systems approach to thermodynamics of stationary states out of equilibrium with attention to the original conceptions which arose at the beginnings of Statistical Mechanics
We discuss here the use of generalized forms of entropy, taken as information measures, to characterize phase transitions and critical behavior in thermodynamic systems. Our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models. The thermodynamic stability of the system is the focus of attention in this geometric context.
Bio-molecules are active agents in that they consume energy to perform tasks. The standard theoretical description, however, considers only a system-external work agent. Fluctuation theorems, for example, do not allow work-exchange between fluctuating molecules. This tradition leaves `action through work', an essential characteristic of an active agent, out of proper thermodynamic consideration. Here, we investigate thermodynamics that considers internal-work. We find a complementary set of relations that capture the production of free energy in molecular interactions while obeying the second …
We consider the stationary state of a Markov process on a bipartite system from the perspective of stochastic thermodynamics. One subsystem is used to extract work from a heat bath while being affected by the second subsystem. We show that the latter allows for a transparent and thermodynamically consistent interpretation of a Maxwell's demon. Moreover, we obtain an integral fluctuation theorem involving the transfer entropy from one subsystem to the other. Comparing three different inequalities, we show that the entropy decrease of the first subsystem provides a tighter bound on the rate of e…
We consider the three-dimensional Ising model in a $L_\perp \times L_\parallel \times L_\parallel$ cuboid geometry with finite aspect ratio $ρ= L_\perp/L_\parallel$ and periodic boundary conditions along all directions. For this model the finite-size scaling functions of the excess free energy and thermodynamic Casimir force are evaluated numerically by means of Monte Carlo simulations. The Monte Carlo results compare well with recent field theoretical results for the Ising universality class at temperatures above and slightly below the bulk critical temperature $T_\mathrm{c}$. Furthermore, th…
Recent single-molecule pulling experiments have shown how it is possible to manipulate RNA molecules using optical tweezers force microscopy. We investigate a minimal model for the experimental setup which includes a RNA molecule connected to two polymers (handles) and a bead, trapped in the optical potential, attached to one of the handles. Initially, we focus on small single-domain RNA molecules which unfold in a cooperative way. The model qualitatively reproduces the experimental results and allow us to investigate the influence of the bead and handles on the unfolding reaction. A main ingr…
We give a thorough Biorthonormal Matrix-Product-State (BMPS) analysis of the Transfer-Matrix Renormalization-Group (TMRG) for non-Hermitian matrices in the thermodynamic limit. The BMPS is built on a dual series of reduced biorthonormal bases for the left and right Perron states of a non-Hermitian matrix. We propose two alternative infinite-size Biorthonormal TMRG (iBTMRG) algorithms and compare their numerical performance in both finite and infinite systems. We show that both iBTMRGs produce a dual infinite-BMPS (iBMPS) which are translationally invariant in the thermodynamic limit. We also d…
We discuss peculiar aspects of the first law of thermodynamics for systems characterized by the presence of meta-equilibrium quasi-stationary states for which the pertinent phase/configuration spaces is generally inhomogeneous. As a consequence, the naive additivity requirement for thermodynamic quantities ceases to be satisfied.
We investigate the first-order transition in the spin-1 two-dimensional Blume-Capel model in square lattices by revisiting the transfer-matrix method. With large strip widths increased up to the size of 18 sites, we construct the detailed phase coexistence curve which shows excellent quantitative agreement with the recent advanced Monte Carlo results. In the deep first-order area, we observe the exponential system-size scaling of the spectral gap of the transfer matrix from which linearly increasing interfacial tension is deduced with decreasing temperature. We find that the first-order signat…
The reliability of physical theories depends on whether they agree with well established physical laws. In this work, we address the compatibility of the Hamiltonian formulation of linear-response theory with the Second Law of Thermodynamics. In order to do so, we verify three complementary aspects often understood as statements of the Second Law: 1. No dissipation for quasistatic process; 2. Dissipation for finite-time processes; 3. Positive entropy production rate. Our analysis focus on two classes of nonequilibrium isothermal processes: slowly-varying and finite-time but weak ones. For the …
We study in great detail the possible existence of a Renyi-associated thermodynamics, with negative results. In particular, we uncover a hidden relation in the Renyi's variational problem (MaxEnt). This relation connects the two associated Lagrange multipliers (Canonical Ensemble) with the mean energy $<U>$ and the Renyi parameter $α$. As a consequence of such relation, we obtain anomalous Renyi-MaxEnt thermodynamic results.
We study the asymptotic behavior of the eigenvalue distribution of the Baxter's corner transfer matrix (CTM) and the density matrix (DM) in the White's density-matrix renormalization group (DMRG), for one-dimensional quantum and two-dimensional classical statistical systems. We utilize the relationship ${\rm DM}={\rm CTM}^4$ which holds for non-critical systems in the thermodynamic limit. Using the known diagonal form of CTM, we derive exact asymptotic form of the DM eigenvalue distribution for the integrable $S=1/2$ XXZ chain (and its related integrable models) in the massive regime. The resu…
It is shown that the equations for pressure, entropy and the isochoric heat capacity obtained by using generalization of the equilibrium thermodynamics in fractional derivatives in the paper mentioned above are approximate, the comparison of the equations with the experimental (tabulated) data for Neon and Argon made in the paper is incorrect, and the conclusions of the paper made on the basis of the comparison could be incorrect. The conditions for validity of the equations are established. It is also established that the question about a physical sense of the exponent of the derivative of a …
Situations where a spontaneous process of energy or matter transfer is enhanced by an external device are widespread in nature (human sweating system, enzyme catalysis, facilitated diffusion across bio-membranes, industrial heat exchangers). The thermodynamics of such processes remains however open. Here we study enhanced heat transfer by a model junction immersed between two thermal baths at different temperatures $T_h$ and $T_c$ ($T_h>T_c$). The transferred heat power is enhanced via controlling the junction by means of external time-dependent fields. Provided that the spontaneous heat flow …
Natural supports of information are given by random copolymers such as DNA or RNA where information is coded in the sequence of covalent bonds. At the molecular scale, the stochastic growth of a single copolymer with or without a template proceeds by successive random attachments or detachments of monomers continuously supplied by the surrounding solution. The thermodynamics of copolymerization shows that fundamental links already exist between information and thermodynamics at the molecular scale, which opens new perspectives to understand the dynamical aspects of information in biology.
The aim of this article is to analyze the relation between the second law of thermodynamics and the so-called arrow of time. For this purpose, a number of different aspects in this arrow of time are distinguished, in particular those of time-(a)symmetry and of (ir)reversibility. Next I review versions of the second law in the work of Carnot, Clausius, Kelvin, Planck, Gibbs, Carathéodory and Lieb and Yngvason, and investigate their connection with these aspects of the arrow of time. It is shown that this connection varies a great deal along with these formulations of the second law. According t…
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and …
We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression. Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that are either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation $Δt\,ΔW \ge (Δt\,ΔW)_{\rm opt}$ between the transfer duration and the expended w…
Entropy might be a not well defined concept if the system can undergo transformations involving stationary nonequilibria. It might be analogous to the heat content (once called ``caloric'') in transformations that are not isochoric (i.e. which involve mechanical work): it could be just a quantity that can be transferred or created, like heat in equilibrium. The text first reviews the philosophy behind a recently proposed definition of entropy production in nonequilibrium stationary systems. A detailed technical attempt at defining the entropy of a stationary states via their variational proper…
How can we derive the evolution equations of dissipative systems? What is the relation between the different approaches? How much do we understand the fundamental aspects of a second law based framework? Is there a hierarchy of dissipative and ideal theories at all? How far can we reach with the new methods of nonequilibrium thermodynamics?
We provide a unified thermodynamic formalism describing information transfers in autonomous as well as nonautonomous systems described by stochastic thermodynamics. We demonstrate how information is continuously generated in an auxiliary system and then transferred to a relevant system that can utilize it to fuel otherwise impossible processes. Indeed, while the joint system satisfies the second law, the entropy balance for the relevant system is modified by an information term related to the mutual information rate between the two systems. We show that many important results previously derive…
At equilibrium, thermodynamic and kinetic information can be extracted from biomolecular energy landscapes by many techniques. However, while static, ensemble techniques yield thermodynamic data, often only dynamic, single-molecule techniques can yield the kinetic data that describes transition-state energy barriers. Here we present a generalized framework based upon dwell-time distributions that can be used to connect such static, ensemble techniques with dynamic, single-molecule techniques, and thus characterize energy landscapes to greater resolutions. We demonstrate the utility of this fra…
Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. Here, the energy cascade is approximated by a combined contribution of a forward drift and diffusion that recover accepted phenomenological theories of turbulence. The fluctuation theorem (FT) is then shown to describe the scale-wise statistics of forward and backward energy transfer and their connection to irreversibility and entropy production. The ensuing turbulence entropy may be used to formulate an extended tur…
In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of pha…
An analytic Bethe ansatz is carried out related to the Lie superalgebra osp(1|2s). We present an eigenvalue formula of a transfer matrix in dressed vacuum form (DVF) labeled by a Young (super) diagram. Remarkable duality among DVFs is found. A complete set of transfer matrix functional relations (T-system) is proposed as a reduction of a Hirota-Miwa equation. We also derive a thermodynamic Bethe ansatz (TBA) equation from this T-system and the quantum transfer matrix method. This TBA equation is identical to the one from the string hypothesis.
