Thermodynamic Entropy Production Research Articles

Thermodynamics of information processing at the molecular scale

The thermodynamics of copolymerization processes leading to the formation and replication of copolymers is presented. Copolymers are natural one-dimensional supports of information, which may be coded in the sequence of monomeric units randomly attached or detached during their synthesis. Multivariate fluctuation relations are here established for copolymerization processes synthesizing Bernoulli and first-order Markov chains. Thereof, the thermodynamic entropy production is deduced and shown to depend on quantities characterizing information possibly stored in the copolymer sequence.

Time reversal invariance, entropy production and work dissipation in stochastic thermodynamics

We consider the work production in a mesosccopic Markov system obeying discrete stochastic dynamics with time-dependent constraints. Using asymmetry relations presented elsewhere, which result from time reversal invariance of the underlying microscopic system, we derive, beside known equalities in stochastic thermodynamics, a new result: the “Carnot equality”, that generalizes the Carnot relation for macroscopic bi-thermal engines. Such equalities, which extend the classical inequalities of thermodynamics, result from microscopic time reversal invariance only. On the other hand we show that, on the mesoscopic level, notions such as entropy production and power dissipation per transition cannot always be defined. In the absence of a precise mechanical model, such definitions are possible if, and only if, the asymmetry relations due to microscopic time reversal invariance are supplemented by space symmetry relations, equivalent to parity, which are not always satisfied. This article is supplemented with comments by J.M.R. Parrondo and L. Granger and a final reply by the authors.

Scattering theory and thermodynamics of quantum transport

Scattering theory is complemented by recent results on full counting statistics, the multivariate fluctuation relation for currents, and time asymmetry in temporal disorder characterized by the Connes‐Narnhofer‐Thirring entropy per unit time, in order to establish relationships with the thermodynamics of quantum transport. Fluctuations in the bosonic or fermionic currents flowing across an open system in contact with particle reservoirs are described by their cumulant generating function, which obeys the multivariate fluctuation relation as the consequence of microreversibility. Time asymmetry in temporal disorder is shown to manifest itself out of equilibrium in the difference between a time‐reversed coentropy and the Connes‐Narnhofer‐Thirring entropy per unit time. This difference is shown to be equal to the thermodynamic entropy production for ideal quantum gases of bosons and fermions. The results are illustrated for a two‐terminal circuit. image

Cycles, randomness, and transport from chaotic dynamics to stochastic processes.

An overview of advances at the frontier between dynamical systems theory and nonequilibrium statistical mechanics is given. Sensitivity to initial conditions is a mechanism at the origin of dynamical randomness-alias temporal disorder-in deterministic dynamical systems. In spatially extended systems, sustaining transport processes, such as diffusion, relationships can be established between the characteristic quantities of dynamical chaos and the transport coefficients, bringing new insight into the second law of thermodynamics. With methods from dynamical systems theory, the microscopic time-reversal symmetry can be shown to be broken at the statistical level of description in nonequilibrium systems. In this way, the thermodynamic entropy production turns out to be related to temporal disorder and its time asymmetry away from equilibrium.

Force–velocity relation for copolymerization processes

A study is reported of copolymerization processes subjected to an external force in the light of recent advances on the kinetics and thermodynamics of copolymer growth. Different processes are considered: the free copolymerization of Bernoulli and first-order Markov chains and copolymerization with a template. For every process, the dependence on the force is analyzed for the elongation rate, the growth velocity, the disorder in the copolymer sequence, the thermodynamic entropy production, as well as related quantities. It is shown that disorder in the copolymer sequence can generate forces of entropic origin. They are characterized by the value of their stall force in the force–velocity relation of the corresponding process.

Scattering approach to the thermodynamics of quantum transport

The thermodynamic entropy production for the scattering processes of noninteracting bosons and fermions in mesoscopic systems is shown to be related to the difference between the Connes–Narnhofer–Thirring entropy per unit time, characterizing temporal disorder in the motion of quantum particles, and the associated time-reversed coentropy per unit time. Under nonequilibrium conditions, the positivity of thermodynamic entropy production can thus be interpreted as a time-reversal symmetry breaking in the temporal disorder of the quantum transport process. Moreover, the full counting statistics of both fermionic and bosonic quantum transport is formulated in relation with the energy and particle currents producing thermodynamic entropy in nonequilibrium steady states.

Efficiency of cellular information processing

We show that a rate of conditional Shannon entropy reduction, characterizing the learning of an internal process about an external process, is bounded by the thermodynamic entropy production. This approach allows for the definition of an informational efficiency that can be used to study cellular information processing. We analyze three models of increasing complexity inspired by the Escherichia coli sensory network, where the external process is an external ligand concentration jumping between two values. We start with a simple model for which ATP must be consumed so that a protein inside the cell can learn about the external concentration. With a second model for a single receptor we show that the rate at which the receptor learns about the external environment can be nonzero even without any dissipation inside the cell since chemical work done by the external process compensates for this learning rate. The third model is more complete, also containing adaptation. For this model we show inter alia that a bacterium in an environment that changes at a very slow time-scale is quite inefficient, dissipating much more than it learns. Using the concept of a coarse-grained learning rate, we show for the model with adaptation that while the activity learns about the external signal the option of changing the methylation level increases the concentration range for which the learning rate is substantial.

Kinetics and thermodynamics of first-order Markov chain copolymerization.

We report a theoretical study of stochastic processes modeling the growth of first-order Markov copolymers, as well as the reversed reaction of depolymerization. These processes are ruled by kinetic equations describing both the attachment and detachment of monomers. Exact solutions are obtained for these kinetic equations in the steady regimes of multicomponent copolymerization and depolymerization. Thermodynamic equilibrium is identified as the state at which the growth velocity is vanishing on average and where detailed balance is satisfied. Away from equilibrium, the analytical expression of the thermodynamic entropy production is deduced in terms of the Shannon disorder per monomer in the copolymer sequence. The Mayo-Lewis equation is recovered in the fully irreversible growth regime. The theory also applies to Bernoullian chains in the case where the attachment and detachment rates only depend on the reacting monomer.

Random paths and current fluctuations in nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical mechanics about the statistics of random paths and current fluctuations. Although statistics is carried out in space for equilibrium statistical mechanics, statistics is considered in time or spacetime for nonequilibrium systems. In this approach, relationships have been established between nonequilibrium properties such as the transport coefficients, the thermodynamic entropy production, or the affinities, and quantities characterizing the microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate. This overview presents results for classical systems in the escape-rate formalism, stochastic processes, and open quantum systems.

On the relationship between dissipation and the rate of spontaneous entropy production from linear irreversible thermodynamics

When systems are far from equilibrium, the temperature, the entropy and the thermodynamic entropy production are not defined and the Gibbs entropy does not provide useful information about the physical properties of a system. Furthermore, far from equilibrium, or if the dissipative field changes in time, the spontaneous entropy production of linear irreversible thermodynamics becomes irrelevant. In 2000 we introduced a definition for the dissipation function and showed that for systems of arbitrary size, arbitrarily near or far from equilibrium, the time integral of the ensemble average of this quantity can never decrease. In the low-field limit, its ensemble average becomes equal to the spontaneous entropy production of linear irreversible thermodynamics. We discuss how these quantities are related and why one should use dissipation rather than entropy or entropy production for non-equilibrium systems.

Limits of Spatial Resolution for Thermography and Other Non-destructive Imaging Methods Based on Diffusion Waves

In this work the measured variable, such as temperature, is a random variable showing fluctuations. The loss of information caused by diffusion waves in non-destructive testing can be described by stochastic processes. In non-destructive imaging, the information about the spatial pattern of a samples interior has to be transferred to the sample surface by certain waves, e.g., thermal waves. At the sample surface these waves can be detected and the interior structure is reconstructed from the measured signals. The amount of information about the interior of the sample, which can be gained from the detected waves on the sample surface, is essentially influenced by the propagation from its excitation to the surface. Diffusion causes entropy production and information loss for the propagating waves. Mandelis has developed a unifying framework for treating diverse diffusion-related periodic phenomena under the global mathematical label of diffusion-wave fields, such as thermal waves. Thermography uses the time-dependent diffusion of heat (either pulsed or modulated periodically) which goes along with entropy production and a loss of information. Several attempts have been made to compensate for this diffusive effect to get a higher resolution for the reconstructed images of the samples interior. In this work it is shown that fluctuations limit this compensation. Therefore, the spatial resolution for non-destructive imaging at a certain depth is also limited by theory.

Distribution of entropy production in a single-electron box

The fluctuation relations are a central concept in thermodynamics at the microscopic scale. These relations are experimentally verified by measuring the entropy production in a single-electron box coupled to two heat baths. Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations1,2,3,4,5 has spurred theoretical6,7,8,9,10,11,12,13 and experimental14,15,16,17,18,19,20,21,22,23,24,25 studies. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, this has not been studied experimentally if there are multiple thermal baths present. Here, we measure, with high precision, the distributions of microscopic entropy production in a single-electron box consisting of two islands with a tunnel junction. The islands are coupled to separate heat baths at different temperatures, maintaining a steady thermal non-equilibrium. We demonstrate that stochastic entropy production8,10,11,12,17,20,25,26 from trajectories of electronic transitions is related to thermodynamic entropy production from dissipated heat in the respective thermal baths. We verify experimentally that the fluctuation relations for both definitions are satisfied. Our results reveal the subtlety of irreversible entropy production in non-equilibrium.

Information-theoretic versus thermodynamic entropy production in autonomous sensory networks

For sensory networks, we determine the rate with which they acquire information about the changing external conditions. Comparing this rate with the thermodynamic entropy production that quantifies the cost of maintaining the network, we find that there is no universal bound restricting the rate of obtaining information to be less than this thermodynamic cost. These results are obtained within a general bipartite model consisting of a stochastically changing environment that affects the instantaneous transition rates within the system. Moreover, they are illustrated with a simple four-states model motivated by cellular sensing. On the technical level, we obtain an upper bound on the rate of mutual information analytically and calculate this rate with a numerical method that estimates the entropy of a time series generated with a simulation.

Irreversible Thermodynamics

We present a maximum-entropy theory of mesoscopic kinetics. The theory gives fully nonlinear nonequilibrium thermodynamic relationships and has no explicit requirement for either microscopic bath variables, an equilibrium energy, or an equilibrium partition function. The entropy maximization process is instead carried out over transition probability distributions with constraints on particle position and velocity updates. The Lagrange multipliers for these constraints express the instantaneous temperature and pressure of external (or microscopic) thermostatic driving systems, with which the distinguished system may or may not eventually reach equilibrium. We show that the analogues of the Gibbs-Maxwell relations and free energy perturbation techniques carry over to fluctuation-dissipation theorems and nonequilibrium ensemble reweighting techniques as should be expected. The result is a fully time-dependent, non-local description of a nonequilibrium ensemble coupled to reservoirs at possibly time-varying thermostatic or mechanical states. We also show that the thermodynamic entropy production extends the generalized fluctuation theorem through the addition of an instantaneous information entropy term for the end-points, leading to a concise statement of the second law of thermodynamics.

Entropy production from stochastic dynamics in discrete full phase space

The stochastic entropy generated during the evolution of a system interacting with an environment may be separated into three components, but only two of these have a non-negative mean. The third component of entropy production is associated with the relaxation of the system probability distribution towards a stationary state and with nonequilibrium constraints within the dynamics that break detailed balance. It exists when at least some of the coordinates of the system phase space change sign under time reversal, and when the stationary state is asymmetric in these coordinates. We illustrate the various components of entropy production, both in detail for particular trajectories and in the mean, using simple systems defined on a discrete phase space of spatial and velocity coordinates. These models capture features of the drift and diffusion of a particle in a physical system, including the processes of injection and removal and the effect of a temperature gradient. The examples demonstrate how entropy production in stochastic thermodynamics depends on the detail that is included in a model of the dynamics of a process. Entropy production from such a perspective is a measure of the failure of such models to meet Loschmidt's expectation of dynamic reversibility.

On the role of damage energy in the fatigue degradation characterization of a composite laminate

The results of a series of experiments to characterize the damage behavior of a glass/epoxy laminate are presented in terms of hysteresis energy dissipation, damage energy variation, energy associated with thermal capacity, temperature rise, and a new method that utilizes thermodynamic entropy production. Results are presented for the observed damage stages and their durations. Different life spans for damage stages are observed based on the plots of damage energy and hysteresis energy. A theoretical model based on the energy balance and the second law of thermodynamics is presented that takes into account the role of hysteresis, damage energy and energy associated with thermal capacity as well as the thermodynamic entropy in the fatigue degradation of a composite laminate. The results of thermodynamic entropy indicate that interpretation of degradation within the laminate requires consideration of both the damage energy and hysteresis energy.

General expression for entropy production in transport processes based on the thermomass model

The entropy production in classical irreversible thermodynamics is expressed as a bilinear form of generalized (driving) forces and conjugated (driven) fluxes, which suffers from the arbitrary decomposition of the forces and the fluxes, and the possible negative entropy production in non-Fourier heat conduction problems (heat waves). This paper presents a general form of the entropy production for heat conduction based on the thermomass model, which is the product of the friction force and the drift velocity of the thermomass divided by the temperature; it holds true for both Fourier and non-Fourier heat conduction. Then a generalization of the entropy production is given for other kinds of linear and nonlinear transport processes. The general expression for entropy production is consistent with that given by extended irreversible thermodynamics, where the system entropy depends not only on the classical variables, but also on the dissipative fluxes, for example, the heat flux in heat conduction problems.

An Information Theory Approach to Nonlinear, Nonequilibrium Thermodynamics

Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complex thermodynamic models by successively adding physical information. We present a new formulation of information algebra that generalizes methods of both information theory and statistical mechanics. From this foundation we derive a theory for ion channel kinetics, identifying a nonequilibrium 'process' free energy functional in addition to the well-known integrated work functionals. The Gibbs-Maxwell relation for the free energy functional is a Green-Kubo relation, applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient thermal and mechanical driving forces. Comparing the physical significance of the Lagrange multipliers to the canonical ensemble suggests definitions of nonequilibrium ensembles at constant capacitance or inductance in addition to constant resistance. Our result is that statistical mechanical descriptions derived from a few primitive algebraic operations on information can be used to create experimentally-relevant and computable models. By construction, these models may use information from more detailed atomistic simulations. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a direct analogue of the second law for thermodynamic entropy production is found by considering information loss in stochastic processes. The information loss identifies a novel contribution from the instantaneous information entropy that ensures non-negative loss.

Quantifying the thermodynamic entropy budget of the land surface: is this useful?

Abstract. As a system is moved away from a state of thermodynamic equilibrium, spatial and temporal heterogeneity is induced. A possible methodology to assess these impacts is to examine the thermodynamic entropy budget and assess the role of entropy production and transfer between the surface and the atmosphere. Here, we adopted this thermodynamic framework to examine the implications of changing vegetation fractional cover on land surface energy exchange processes using the NOAH land surface model and eddy covariance observations. Simulations that varied the relative fraction of vegetation were used to calculate the resultant entropy budget as a function of fraction of vegetation. Results showed that increasing vegetation fraction increases entropy production by the land surface while decreasing the overall entropy budget (the rate of change in entropy at the surface). This is accomplished largely via simultaneous increase in the entropy production associated with the absorption of solar radiation and a decline in the Bowen ratio (ratio of sensible to latent heat flux), which leads to increasing the entropy export associated with the latent heat flux during the daylight hours and dominated by entropy transfer associated with sensible heat and soil heat fluxes during the nighttime hours. Eddy covariance observations also show that the entropy production has a consistent sensitivity to land cover, while the overall entropy budget appears most related to the net radiation at the surface, however with a large variance. This implies that quantifying the thermodynamic entropy budget and entropy production is a useful metric for assessing biosphere-atmosphere-hydrosphere system interactions.

Fluctuation theorems and atypical trajectories

In this work, we have studied simple models that can be solved analytically to illustrate various fluctuation theorems. These fluctuation theorems provide symmetries individually to the distributions of physical quantities such as the classical work (Wc), thermodynamic work (W), total entropy (Δstot) and dissipated heat (Q), when the system is driven arbitrarily out of equilibrium. All these quantities can be defined for individual trajectories. We have studied the number of trajectories which exhibit behaviour unexpected at the macroscopic level. As the time of observation increases, the fraction of such atypical trajectories decreases, as expected at the macroscale. The distributions for the thermodynamic work and entropy production in nonlinear models may exhibit a peak (most probable value) in the atypical regime without violating the expected average behaviour. However, dissipated heat and classical work exhibit a peak in the regime of typical behaviour only.