arXiv subjects

Gatien Verley

Publications and source records attributed to Gatien Verley.

At least 19 recordsLinked to original sources

arXiv2024

Thermodynamic Circuits 2: Nonequilibrium conductance matrix for a thermoelectric converter

In the linear regime, Onsager's response matrix provides the coupling between heat and charge currents crossing a section of thermoelectric materials of infinitesimal thickness. Integrating this response over the finite thickness of a one-dimensional Thermoelectric Converter (TEC) leads to quadratic heat-force characteristics (Joule's law) and linear current-voltage characteristics (Ohm's law). However, these non-linear characteristic equations are not matrix relation anymore. This prevents from determining the currents degree of coupling, albeit its central role for optimizing energy conversion. Based on current conservation laws, i.e., the linear dependence between internal physical currents (crossing a section of material) or between external ones (exchanged with the environment), we distinguish two relevant basis of physical and fundamental currents. For those, we define non-equilibrium conductance matrices providing the current-force relations of a TEC in any convenient basis. In doing so, we introduce a degree of coupling between heat and charge currents, in line with the work of Kedem and Caplan but beyond weakly irreversible thermodynamics. This demonstrates by example that non-equilibrium conductance matrices constitute effective models for driven systems, as Onsager response matrices do in the linear regime. The sequel papers of this series focus on associating systems modeled in such way.

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arXiv2024

Thermodynamic Circuits: Modeling chemical reaction networks with nonequilibrium conductance matrices

We derive the nonequilibrium conductance matrix for open stationary Chemical Reaction Networks (CRNs) described by a deterministic mass action kinetic equation. As an illustration, we determine the nonequilibrium conductance matrix of a CRN made of two pseudo-linear sub-networks, called chemical modules, in two different ways: First by computing the nonequilibrium conductances of the modules that are then serially connected. Second by computing the nonequilibrium conductance of the CRN directly. The two approaches coincide, as expected from our theory of thermodynamic circuits.

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arXiv2024

Thermodynamic Circuits: Association of thermoelectric converters in stationary non-equilibrium

Following up on the recently published circuit theory for thermodynamic devices, we consider networks of Thermo-Electric Converters (TECs) in stationary non-equilibrium. Assuming constant thermoelectric properties, the integration over a finite thickness of the linear local response of the thermoelectric material yields the non-linear current-force characteristics. We show how to derive a choice of nonequilibrium conductance matrix summarizing the current-force characteristics for every available sets of currents and forces. This problem has infinitely many solutions if one considers only thermodynamic constraints. Each solution differs, among others, by the coupling between the currents. Then, we determine the current-force characteristics of the serial (respectively parallel) association of two TECs using the laws of resistance (respectively conductance) matrix addition. For TECs in series, we find current-dependent boundary conditions for each sub-device. Since currents derive from composite potentials, we also associate the derivability and continuity of these potentials at the interfaces with conditions on thermoelectric coefficients. For TECs in parallel, we discuss the possibility of loop currents that are forbidden for the serial association.

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arXiv2023

Thermodynamic Circuits I: Association of devices in stationary nonequilibrium

For a circuit made of thermodynamic devices in stationary nonequilibrium, we determine the mean currents (of energy, matter, charge, etc) exchanged with external reservoirs driving the circuit out of equilibrium. Starting from the conductance matrix describing the nonlinear current--force characteristics of each device, we obtain the conductance matrix of the composite device. This generalizes the rule of resistance addition (serial association) or conductance addition (parallel association) in stationary out-of-equilibrium thermodynamics and for multiple coupled potentials and currents of different natures. Our work emphasizes the pivotal role of conservation laws when creating circuits of complex devices. Finally, two examples illustrate the determination of the conservation laws for the serial and parallel associations of thermodynamic devices.

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arXiv2021

Dynamical equivalence classes for Markov jump processes

Two different Markov jump processes driven out of equilibrium by constant thermodynamic forces may have identical current fluctuations in the stationary state. The concept of dynamical equivalence classes emerges from this statement as proposed by Andrieux for discrete-time Markov chains on simple graphs. We define dynamical equivalence classes in the context of continuous-time Markov chains on multigraphs using the symmetric part of the rate matrices that define the dynamics. The freedom on the skew-symmetric part is at the core of the freedom inside a dynamical equivalence class. It arises from different splittings of the thermodynamic forces onto the system's transitions.

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arXiv2021

Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume

When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker-Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron-Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian-Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton-Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron-Frobenius theorem concerning the positivity of the dominant eigenvector. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.

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arXiv2020

Efficiency fluctuations of stochastic machines undergoing a phase transition

We study the efficiency fluctuations of a stochastic heat engine made of $N$ interacting unicyclic machines and undergoing a phase transition in the macroscopic limit. Depending on $N$ and on the observation time, the machine can explore its whole phase space or not. This affects the engine efficiency that either strongly fluctuates on a large interval of equiprobable efficiencies (ergodic case) or fluctuates close to several most likely values (nonergodic case). We also provide a proof that despite the phase transition, the decay rate of the efficiency distribution at the reversible efficiency remains largest one although other efficiencies can now decay equally fast.

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arXiv2019

Periodically driven jump processes conditioned on large deviations

We study the fluctuations of systems modeled by Markov jump processes with periodic generators. We focus on observables defined through time-periodic functions of the system's states or transitions. Using large deviation theory, canonical biasing and generalized Doob transform, we characterize the asymptotic fluctuations of such observables after a large number of periods by obtaining the Markov process that produces them. We show that this process, called driven process, is the minimum under constraint of the large deviation function for occupation and jumps.

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arXiv2019

An ordered set of power-efficiency trade-offs

In this paper, we derive a number of inequalities which express power-efficiency trade-offs that hold generally for thermodynamic machines operating in non-equilibrium stationary states. One of these inequalities concerns the output power, which is bounded by a quadratic function of the thermodynamic efficiency multiplied by a factor. Different factors can be obtained according to the level of knowledge one has about the underlying dynamics of the machine, they can depend for instance on the covariance of the input flux, the dynamical activity, or the non-equilibrium conductance.

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arXiv2018

Non equivalence of dynamical ensembles and emergent non ergodicity

Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.

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arXiv2018

Degree of coupling and efficiency of energy converters far-from-equilibrium

In this paper, we introduce a real symmetric and positive semi-definite matrix, which we call the non-equilibrium conductance matrix, and which generalizes the Onsager response matrix for a system in a non-equilibrium stationary state. We then express the thermodynamic efficiency in terms of the coefficients of this matrix using a parametrization similar to the one used near equilibrium. This framework, then valid arbitrarily far from equilibrium allows to set bounds on the thermodynamic efficiency by a universal function depending only on the degree of coupling between input and output currents. It also leads to new general power-efficiency trade-offs valid for macroscopic machines that are compared to trade-offs previously obtained from uncertainty relations. We illustrate our results on an unicycle heat to heat converter and on a discrete model of molecular motor.

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arXiv2017

Collective effects enhancing power and efficiency

Energy conversion is most efficient for micro or nano machines with tight coupling between input and output power. To reach meaningful amounts of power, ensembles of $N$ such machines must be considered. We use a model system to demonstrate that interactions between $N$ tightly coupled nanomachines can enhance the power output per machine. Furthermore, while interactions break tight coupling and thus lower efficiency in finite ensembles, the macroscopic limit ($N \rightarrow \infty$) restores it and enhances both the efficiency and the output power per nanomachine.

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arXiv2016

Efficiency fluctuations of small machines with unknown losses

The efficiency statistics of a small thermodynamic machine has been recently investigated assuming that the total dissipation was a linear combination of two currents: the input and output currents. Here, we relax this standard assumption and reconsider the question of the efficiency fluctuations for a machine involving three different processes, first in full generality and second for two different examples. Since the third process may not be measurable and/or may decrease the machine efficiency, our motivation is to study the effect of unknown losses in small machines.

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arXiv2016

Nonequilibrium thermodynamic potentials for continuous-time Markov chains

We connect the rare fluctuations of an Equilibrium (EQ) process and the typical fluctuations of a nonequilibrium (NE) stationary process. In the framework of large deviation theory, this observation allows us to introduce NE thermodynamic potentials. For continuous-time Markov chains, we identify the relevant pairs of conjugated variables and propose two NE ensembles: one with fixed dynamics and fluctuating time-averaged variables, and another with fixed time-averaged variables, but a fluctuating dynamics. Accordingly, we show that NE processes are equivalent to conditioned EQ processes ensuring that NE potentials are Legendre dual. We find a variational principle satisfied by the NE potentials that reach their maximum in the NE stationary state and whose first derivatives produce the NE equations of state, and second derivatives produce the NE Maxwell relations generalizing the Onsager reciprocity relations.

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arXiv2015

Efficiency statistics at all times: Carnot limit at finite power

We derive the statistics of the efficiency under the assumption that thermodynamic fluxes fluctuate with normal law, parametrizing it in terms of time, macroscopic efficiency, and a coupling parameter $ζ$. It has a peculiar behavior: No moments, one sub- and one super-Carnot maxima corresponding to reverse operating regimes (engine/pump), the most probable efficiency decreasing in time. The limit $ζ\to 0$ where the Carnot bound can be saturated gives rise to two extreme situations, one where the machine works at its macroscopic efficiency, with Carnot limit corresponding to no entropy production, and one where for a transient time scaling like $1/ζ$ microscopic fluctuations are enhanced in such a way that the most probable efficiency approaches Carnot at finite entropy production.

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arXiv2014

Universal theory of efficiency fluctuations

Using the fluctuation theorem supplemented with geometric arguments, we derive universal features of the (long-time) efficiency fluctuations for thermal and isothermal machines operating under steady or periodic driving, close or far from equilibrium. In particular, the long-time probability for observing a reversible efficiency in a given engine is identical to that for the same engine working under the time-reversed driving. When the driving is symmetric, this reversible efficiency becomes the least probable.

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arXiv2014

The unlikely Carnot efficiency

The efficiency of an heat engine is traditionally defined as the ratio of its average output work over its average input heat. Its highest possible value was discovered by Carnot in 1824 and is a cornerstone concept in thermodynamics. It led to the discovery of the second law and to the definition of the Kelvin temperature scale. Small-scale engines operate in the presence of highly fluctuating input and output energy fluxes. They are therefore much better characterized by fluctuating efficiencies. In this study, using the fluctuation theorem, we identify universal features of efficiency fluctuations. While the standard thermodynamic efficiency is, as expected, the most likely value, we find that the Carnot efficiency is, surprisingly, the least likely in the long time limit. Furthermore the probability distribution for the efficiency assumes a universal scaling form when operating close-to-equilibrium. We illustrate our results analytically and numerically on two model systems.

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arXiv2014

Work statistics in stochastically driven systems

We identify the conditions under which a stochastic driving inducing energy changes on a system coupled to a thermal bath can be treated as a work source. When these conditions are met, the work statistics satisfies the Crooks fluctuation theorem traditionally derived for deterministic drivings. We illustrate this fact by calculating and comparing the work statistics for a two-level system driven respectively by a stochastic and a deterministic piecewise constant protocol.

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