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Employing a recent technology of tree surgery we prove a ``deletion-constriction'' formula for products of rooted spanning trees on weighted directed graphs that generalizes deletion-contraction on undirected graphs. The formula implies that, letting $\tau_x^\varnothing$, $\tau_x^+$, and $\tau_x^-$ be the rooted spanning tree polynomials obtained respectively by removing an edge in both directions or by forcing the tree to pass through either direction of that edge, the vectors $(\tau_x^\varnothing, \tau_x^+, \tau_x^-)$ are coplanar for all roots $x$. We deploy the result to give an alternative derivation of a recently found mutual linearity of stationary currents of Markov chains. We generalize deletion-constriction and current linearity among two edges, and conjecture that similar results may hold for arbitrary subsets of edges., Comment: 21 pages, 3 figures

For continuous-time Markov chains and open unimolecular chemical reaction networks, we prove that any two stationary currents are linearly related upon perturbations of a single edge's transition rates, arbitrarily far from equilibrium. We extend the result to non-stationary currents in the frequency domain, provide and discuss an explicit expression for the current-current susceptibility in terms of the network topology, and discuss possible generalizations. In practical scenarios, the mutual linearity relation has predictive power and can be used as a tool for inference or model proof-testing., Comment: Accepted for publication in Physical Review Letters

Coarse-grained models are widely used to explain the effective behavior of partially observable physical systems with hidden degrees of freedom. Reduction procedures in state space typically disrupt Markovianity and a fluctuation relation cannot be formulated. A recently developed framework of transition-based coarse-graining gave rise to a fluctuation relation for a single current, while all others are hidden. Here, we extend the treatment to an arbitrary number of observable currents. Crucial for the derivation are the concepts of mixed currents and their conjugated effective affinities, that can be inferred from the time series of observable transitions. We also discuss the connection to generating functions, transient behavior, and how our result recovers the fluctuation relation for a complete set of currents., Comment: 37 pages, 11 figures

Lecture notes after the doctoral school (Post)Modern Thermodynamics held at the University of Luxembourg, December 2022, 5-7, covering and advancing continuous-time Markov chains, network theory, stochastic thermodynamics, large deviations, deterministic and stochastic chemical reaction networks, metastability, martingales, quantum thermodynamics, and foundational issues., Comment: Submission to SciPost

Stochastic chemical processes are described by the chemical master equation satisfying the law of mass-action. We first ask whether the dual master equation, which has the same steady state as the chemical master equation, but with inverted reaction currents, satisfies the law of mass-action, namely, still describes a chemical process. We prove that the answer depends on the topological property of the underlying chemical reaction network known as deficiency. The answer is yes only for deficiency-zero networks. It is no for all other networks, implying that their steady-state currents cannot be inverted by controlling the kinetic constants of the reactions. Hence, the network deficiency imposes a form of non-invertibility to the chemical dynamics. We then ask whether catalytic chemical networks are deficiency-zero. We prove that the answer is no when they are driven out of equilibrium due to the exchange of some species with the environment.

For continuous-time Markov chains we prove that, depending on the notion of effective affinity $F$, the probability of an edge current to ever become negative is either $1$ if $F< 0$ else $\sim \exp - F$. The result generalizes a ``noria'' formula to multicyclic networks. We give operational insights on the effective affinity and compare several estimators, arguing that stopping problems may be more accurate in assessing the nonequilibrium nature of a system according to a local observer. Finally we elaborate on the similarity with the Boltzmann formula. The results are based on a constructive first-transition approach., Comment: 25 pages, 2 figures

The infimum of an integrated current is its extreme value against the direction of its average flow. Using martingale theory, we show that the infima of integrated edge currents in time-homogeneous Markov jump processes are geometrically distributed, with a mean value determined by the effective affinity measured by a marginal observer that only sees the integrated edge current. In addition, we show that a marginal observer can estimate a finite fraction of the average entropy production rate in the underlying nonequilibrium process from the extreme value statistics in the integrated edge current. The estimated average rate of dissipation obtained in this way equals the above mentioned effective affinity times the average edge current. Moreover, we show that estimates of dissipation based on extreme value statistics can be significantly more accurate than those based on thermodynamic uncertainty ratios, as well as those based on a naive estimator obtained by neglecting nonMarkovian correlations in the Kullback-Leibler divergence of the trajectories of the integrated edge current., Comment: 38 pages, 6 figures

The modern thermodynamics of discrete systems is based on graph theory, which provides both algebraic methods to define observables and a geometric intuition of their meaning and role. However, because chemical reactions are usually many-to-many, chemical networks are rather described by hypergraphs, which lack a systematized algebraic treatment and a clear geometric intuition. Here we fill this gap by building fundamental bases of chemical cycles (encoding stationary behavior) and cocycles (encoding finite-time relaxation). We interpret them in terms of circulations and gradients on the hypergraph, and use them to properly identify nonequilibrium observables. As application, we unveil hidden symmetries in linear response and, within this regime, propose a reconstruction algorithm for large metabolic networks consistent with Kirchhoff's Voltage and Current Laws., Comment: 36 pages, 12 figures, 1 table; an appendix has been added

The fluctuation relation, a milestone of modern thermodynamics, is only established when a set of fundamental currents can be measured. Here we prove that it also holds for systems with hidden transitions if observations are carried ``at their own beat,'' that is, by stopping the experiment after a fixed number of visible transitions, rather than the elapse of an external clock time. This suggests that thermodynamic symmetries are more resistant to the loss of information when described in the space of transitions., Comment: 10 pages, 7 figures

Interpreting partial information collected from systems subject to noise is a key problem across scientific disciplines. Theoretical frameworks often focus on the dynamics of variables that result from coarse-graining the internal states of a physical system. However, most experimental apparatuses can only detect a partial set of transitions, while internal states are inaccessible. Here, we consider an observer who records a time series of occurrences of one or several transitions performed by a system, under the assumption that its underlying dynamics is Markovian. We pose the question of how one can use the transitions' information to make inferences of dynamical, thermodynamical, and biochemical properties. First, elaborating on first-passage time techniques, we derive analytical expressions for the probabilities of consecutive transitions and the time elapsed between them. Second, we derive a lower bound for the entropy production rate that equals the sum of two non-negative contributions, one due to the statistics of transitions and a second due to the statistics of inter-transition times. We also show that when only one current is measured, our estimate still detects irreversibility even in the absence of net currents. We illustrate the developed framework in experimentally-validated biophysical models of kinesin and dynein molecular motors, and in a minimal model for template-directed polymerization. Our results reveal that while entropy production is entailed in the statistics of two successive transitions of the same type, the statistics of two different successive transitions can probe the existence of an underlying disorder in the motion of a molecular motor. Taken all together, our results highlight the power of inference from transition statistics ranging from thermodynamic quantities to network-topology properties of Markov processes.

Several recent inequalities bound the precision of a current - counting net number of transitions in a system - by a thermodynamic measure of dissipation. However, while currents may be defined locally, dissipation is a global property. Inspired by the fact that ever since Carnot cycles are the unit elements of thermodynamic processes, we prove similar bounds tailored to cycle currents - counting net cycle completions - in terms of their conjugate affinities. We show that these inequalities are stricter than previous ones, even far from equilibrium, and that they allow to tighten those on transition currents. We illustrate our results with a simple model and discuss some technical and conceptual issues related to shifting attention from transition to cycle observables., Comment: 5 pages, 3 figures

We derive an expression for the minimal rate of entropy that sustains two reservoirs at different temperatures $T_0$ and $T_\ell$. The law displays an intuitive $\ell^{-1}$ dependency on the relative distance and a characterisic $\log^2 (T_\ell/T_0)$ dependency on the boundary temperatures. First we give a back-of-envelope argument based on the Fourier Law (FL) of conduction, showing that the least-dissipation profile is exponential. Then we revisit a model of a chain of oscillators, each coupled to a heat reservoir. In the limit of large damping we reobtain the exponential and squared-log behaviors, providing a self-consistent derivation of the FL. For small damping "equipartition frustration" leads to a well-known balistic behaviour, whose incompatibility with the FL posed a long-time challenge., Comment: Re-Submission to SciPost

Open systems may be perturbed out of equilibrium states either by subjecting them to nonconservative forces or by injecting external currents. For small perturbations, the linear response is quantified by two different matrices. In the framework of network thermodynamics, under very broad assumptions, we show that the two matrices are connected by a supersymmetry, predicting that they have the same spectrum - up to the degeneracy of the ground state. Our approach brings into the mathematics of supersymmetry a new ingredient, namely oblique projection operators., Comment: 5 pages, 2 figures

The response of thermodynamic systems perturbed out of an equilibrium steady-state is described by the reciprocal and the fluctuation-dissipation relations. The so-called fluctuation theorems extended the study of fluctuations far beyond equilibrium. All these results rely on the crucial assumption that the observer has complete information about the system. Such a precise control is difficult to attain, hence the following questions are compelling: Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only establish local equilibrium amidst the whirling of hidden degrees of freedom, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations? We model the dynamics of open systems as Markov jump processes on finite networks. We establish that: 1) While marginal currents do not obey a full-fledged fluctuation relation, there exist effective affinities for which an integral fluctuation relation holds; 2) Under reasonable assumptions on the parametrization of the rates, effective and "real" affinities only differ by a constant; 3) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds while reciprocity does not; 4) There exists a notion of marginal time-reversal that plays a role akin to that played by time-reversal for complete systems, which restores the fluctuation relation and reciprocity. The above results hold for configuration-space currents, and for phenomenological currents provided that certain symmetries of the effective affinities are respected - a condition whose range of validity we deem the most interesting question left open to future inquiry. Our results are constructive and operational: we provide an explicit expression for the effective affinities and propose a procedure to measure them in laboratory., Comment: 41 pages. Comments are welcomed

It is known that complementary oblique projections $\hat{P}_0 + \hat{P}_1 = I$ on a Hilbert space $\mathscr{H}$ have the same standard operator norm $\|\hat{P}_0\| = \|\hat{P}_1\|$ and the same singular values, but for the multiplicity of $0$ and $1$. We generalize these results to Hilbert spaces endowed with a positive-definite metric $G$ on top of the scalar product. Our main result is that the volume elements (pseudodeterminants $\det_+$) of the metrics $L_0,L_1$ induced by $G$ on the complementary oblique subspaces $\mathscr{H} = \mathscr{H}_0 \oplus \mathscr{H}_1$, and of those $\mathit{\Gamma}_0,\mathit{\Gamma}_1$ induced on their algebraic duals, obey the relations \begin{align} \frac{\det_+ L_1}{\det_+ \mathit{\Gamma}_0} = \frac{\det_+ L_0}{\det_+ \mathit{\Gamma}_1} = {\det}_+ G. \nonumber \end{align} Furthermore, we break this result down to eigenvalues, proving a "supersymmetry" of the two operators $\sqrt{\mathit{\Gamma}_0 L_0}$ and $\sqrt{L_1 \mathit{\Gamma}_1}$. We connect the former result to a well-known duality property of the weighted-spanning-tree polynomials in graph theory., Comment: 11 pages

Systems driven away from thermal equilibrium constantly deliver entropy to their environment. Determining this entropy production requires detailed information about the system's internal states and dynamics. However, in most practical scenarios, only a part of a complex experimental system is accessible to an external observer. In order to address this challenge, two notions of partial entropy production have been introduced in the literature as a way to assign an entropy production to an observed subsystem: one due to Shiraishi and Sagawa [Phys. Rev. E 91, 012130 (2015)] and another due to Polettini and Esposito [arXiv:1703.05715 (2017)]. We show that although both of these schemes provide a lower bound on the total entropy production, the latter -- which utilizes an effective thermodynamics description-- gives a better estimate of the total dissipation. Using this effective thermodynamic framework, we establish a partitioning of the total entropy production into two contributions that individually verify integral fluctuation theorems: an observable partial entropy production and a hidden entropy production assigned to the unobserved subsystem. Our results offer broad implications for both theoretical and empirical systems when only partial information is available.

Thermodynamics is usually formulated on the presumption that the observer has complete information about the system he/she deals with: no parasitic current, exact evaluation of the forces that drive the system. For example, the acclaimed Fluctuation Relation (FR), relating the probability of time-forward and time-reversed trajectories, assumes that the measurable transitions suffice to characterize the process as Markovian (in our case, a continuous-time jump process). However, most often the observer only measures a marginal current. We show that he/she will nonetheless produce an effective description that does not dispense with the fundamentals of thermodynamics, including the FR and the 2nd law. Our results stand on the mathematical construction of a hidden time reversal of the dynamics, and on the physical requirement that the observed current only accounts for a single transition in the configuration space of the system. We employ a simple abstract example to illustrate our results and to discuss the feasibility of generalizations., Comment: 8 pages, 1 figure

The widely debated feasibility of thermodynamic machines achieving Carnot efficiency at finite power has been convincingly dismissed. Yet, the common wisdom that efficiency can only be optimal in the limit of infinitely-slow processes overlooks the dual scenario of infinitely-fast processes. We corroborate that efficient engines at divergent power output are not theoretically impossible, framing our claims within the theory of Stochastic Thermodynamics. We inspect the case of an electronic quantum dot coupled to three particle reservoirs to illustrate the physical rationale., Comment: 5 pages, 2 figures

Near equilibrium, where all currents of a system vanish on average, the fluctuation-dissipation relation (FDR) connects a current's spontaneous fluctuations with its response to perturbations of the conjugate thermodynamic force. Out of equilibrium, fluctuation-response relations generally involve additional nondissipative contributions. Here, in the framework of stochastic thermodynamics, we show that an equilibrium-like FDR holds for internally equilibrated currents, if the perturbing conjugate force only affects the microscopic transitions that contribute to the current. We discuss the physical requirements for the validity of our result and apply it to nano-sized electronic devices., Comment: In this revised version of our manuscript, we generalized our result and improved the presentation in order to emphasize its physical significance. The mathematical details of the proof of our main results have been moved to a supplementary material. Keywords: nonequilibrium & irreversible thermodynamics; fluctuation-dissipation theorems

Phenomenological nonequilibrium thermodynamics describes how fluxes of conserved quantities such as matter, energy and charge flow from outer reservoirs across a system, and how they irreversibly degrade from one form to another. Stochastic thermodynamics is formulated in terms of probability fluxes circulating in the system's configuration space. The consistency of the two frameworks is granted by the condition of local detailed balance, which specifies the amount of physical quantities exchanged with the reservoirs during single transitions between configurations. We demonstrate that the topology of the configuration space crucially determines the number of independent thermodynamic affinities (forces) that the reservoirs generate across the system, and provide a general algorithm that produces the fundamental affinities and their conjugate currents contributing to the total dissipation, based on the interplay between macroscopic conservations laws for the currents and microscopic symmetries of the affinities., Comment: 5 pages, 1 figure

We study the effect of intrinsic noise on the thermodynamic balance of complex chemical networks subtending cellular metabolism and gene regulation. A topological network property called deficiency, known to determine the possibility of complex behavior such as multistability and oscillations, is shown to also characterize the entropic balance. In particular, only when deficiency is zero does the average stochastic dissipation rate equal that of the corresponding deterministic model, where correlations are disregarded. In fact, dissipation can be reduced by the effect of noise, as occurs in a toy model of metabolism that we employ to illustrate our findings. This phenomenon highlights that there is a close interplay between deficiency and the activation of new dissipative pathways at low molecule numbers., Comment: 10 Pages, 6 figures

We provide an exact expression for the statistics of the fluxes of Markov jump processes at all times, improving on asymptotic results from large deviation theory. The main ingredient is a generalization of the BEST theorem in enumeratoric graph theory to Eulerian tours with open ends. In the long-time limit we reobtain Sanov's theorem for Markov processes, which expresses the exponential suppression of fluctuations in terms of relative entropy. The finite-time power-law term, increasingly important with the system size, is a spanning-tree determinant that, by introducing Grassmann variables, can be absorbed into the effective Lagrangian of a Fermionic ghost field on a metric space, coupled to a gauge potential. With reference to concepts in nonequilibrium stochastic thermodynamics, the metric is related to the dynamical activity that measures net communication between states, and the connection is made to a previous gauge theory for diffusion processes., Comment: 22 pages

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 $\zeta$. 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 $\zeta\to 0$ 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 $1/\zeta$ microscopic fluctuations are enhanced in such a way that the most probable efficiency approaches Carnot at finite entropy production., Comment: 5+4 pages, 4 figures. Title modified

We introduce the Fisher information in the basis of decay modes of Markovian dynamics, arguing that it encodes important information about the behavior of nonequilibrium systems. In particular we generalize an orthonormality relation between decay eigenmodes of detailed balanced systems to normal generators that commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical distributions, we relate the result to the choice of a coordinate patch that makes the Fisher-Rao metric Euclidean at the steady distribution, realizing a sort of statistical equivalence principle. We then classify nonequilibrium systems according to their spectrum, showing that a degenerate Fisher matrix is the signature of the insurgence of a class of dynamical phase transitions between nonequilibrium regimes, characterized by level crossing and power-law decay in time of suitable order parameters. An important consequence is that normal systems cannot manifest critical behavior. Finally, we study the Fisher matrix of systems with time-scale separation., Comment: 9 pages, 1 figure. Edited version of arXiv:1105.4134

We prove a transient fluctuation theorem for the currents for continuous-time Markov jump processes with stationary rates, generalizing an asymptotic result by Andrieux and Gaspard [J. Stat. Phys. 127, 107 (2007)] to finite times. The result is based on a graph theoretical decomposition in cycle currents and an additional set of tidal currents that characterize the transient relaxation regime. The tidal term can then be removed by a preferred choice of a suitable initial equilibrium ensemble, a result that provides the general theory for the fluctuation theorem without ensemble quantities recently addressed in [Phys. Rev. E 89, 052119 (2014)]. As an example we study the reaction network of a simple stochastic chemical engine, and finally we digress on general properties of fluctuation relations for more complex chemical reaction networks., Comment: 19 pages, 2 figures. Sign error corrected in Eq.(50) and following

It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges to an arbitrarily chosen spanning tree. In this paper we show that the edge vector space can also be decomposed in terms of cycles and the generating edges of cocycles (called cochords), or of cocycles and the generating edges of cycles (called chords). From this observation follows a construction in terms of oblique complementary projection operators. We employ this algebraic construction to prove several properties of unweighted Kirchhoff-Symanzik matrices, encoding the mutual superposition between cycles and cocycles. In particular, we prove that dual matrices of planar graphs have the same spectrum (up to multiplicities). We briefly comment on how this construction provides a refined formalization of Kirchhoff's mesh analysis of electrical circuits, which has lately been applied to generic thermodynamic networks.

In this and a companion paper we outline a general framework for the thermodynamic description of open chemical reaction networks, with special regard to metabolic networks regulating cellular physiology and biochemical functions. We first introduce closed networks "in a box", whose thermodynamics is subjected to strict physical constraints: the mass-action law, elementarity of processes, and detailed balance. We further digress on the role of solvents and on the seemingly unacknowledged property of network independence of free energy landscapes. We then open the system by assuming that the concentrations of certain substrate species (the chemostats) are fixed, whether because promptly regulated by the environment via contact with reservoirs, or because nearly constant in a time window. As a result, the system is driven out of equilibrium. A rich algebraic and topological structure ensues in the network of internal species: Emergent irreversible cycles are associated to nonvanishing affinities, whose symmetries are dictated by the breakage of conservation laws. These central results are resumed in the relation $a + b = s^Y$ between the number of fundamental affinities $a$, that of broken conservation laws $b$ and the number of chemostats $s^Y$. We decompose the steady state entropy production rate in terms of fundamental fluxes and affinities in the spirit of Schnakenberg's theory of network thermodynamics, paving the way for the forthcoming treatment of the linear regime, of efficiency and tight coupling, of free energy transduction and of thermodynamic constraints for network reconstruction., Comment: 18 pages

"Ceci n'est pas une pipe" wrote Ren\'e Magritte on what was only the representation of a pipe. Phenomena and their physical descriptions differ, and in particular the laws ruling the former might enjoy symmetries that have to be spent to attain the latter. So, inertial frames are necessary to draw numbers out of Newtonian mechanics and confront with experiment, but ultimately the laws of mechanics are independent of reference frames. Generalizing work done in Ref. [M. Polettini, EPL 97 (2012) 30003] to continuous systems, we discuss from a foundational point of view how subjectivity in the choice of reference prior probability is a (gauge) symmetry of thermodynamics. In particular, a change of priors corresponds to a change of coordinates. Employing an approach based on the stochastic thermodynamics of continuous state-space diffusion processes, we discuss the difference between thermostatic and thermodynamic observables and show that, while the quantification of entropy depends on priors, the second law of thermodynamics is formulated in terms of invariant quantities, in particular the curvature of the thermodynamic force (gauge potential), which we calculate in a few examples of processes led by different nonequilibrium mechanisms., Comment: 6 pages

We challenge claims that the principle of maximum entropy production produces physical phenomenological relations between conjugate currents and forces, even beyond the linear regime, and that currents in networks arrange themselves to maximize entropy production as the system approaches the steady state. In particular: (1) we show that Ziegler's principle of thermodynamic orthogonality leads to stringent reciprocal relations for higher order response coefficients, and in the framework of stochastic thermodynamics, we exhibit a simple explicit model that does not satisfy them; (2) on a network, enforcing Kirchhoff's current law, we show that maximization of the entropy production prescribes reciprocal relations between coarse-grained observables, but is not responsible for the onset of the steady state, which is rather due to the minimum entropy production principle., Comment: 21 pages

We show via counterexamples that relative entropy between the solution of a Markovian master equation and the steady state is not a convex function of time. We thus let down a curtain on a possible formulation of a principle of thermodynamics regarding decrease of the nonadiabatic entropy production. However, we argue that a large separation of typical decay times is necessary for nonconvex solutions to occur, making concave transients extremely short-lived with respect to the main relaxation modes. We describe a general method based on the Fisher information matrix to discriminate between generators that do and don't admit nonconvex solutions. While initial conditions leading to concave transients are shown to be extremely fine-tuned, by our method we are able to select nonconvex initial conditions that are arbitrarily close to the steady state. Convexity does occur when the system is close to satisfy detailed balance, or more generally when certain normality conditions of the decay modes are satisfied. Our results circumscribe the range of validity of a conjecture proposed by Maes et al. [Phys. Rev. Lett. 107, 010601 (2011)] regarding monotonicity of the large deviation rate functional for the occupation probability (dynamical activity), showing that while the conjecture might still hold in the long time limit, the dynamical activity is not a Lyapunov function., Comment: 11 pages, 3 figures

We propose a Langevin equation for systems in an environment with nonuniform temperature. At odds with an older proposal, ours admits a locally Maxwellian steady state, local equipartition holds and for detailed-balanced (reversible) systems statistical and physical entropies coincide. We describe its thermodynamics, which entails a generalized version of the First Law and Clausius's characterization of reversibility. Finally, we show that a Brownian particle constrained into a smooth curve behaves according to our equation, as if experiencing nonuniform temperature.

Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of non gauge-invariant systems is not unambiguously defined. They typically do not relax to equilibrium steady states even in the absence of extenal forces. Assuming both coordinate covariance and gauge invariance, we derive a second-order Langevin equation with state-dependent diffusion matrix and vanishing environmental forces. It differs from previous proposals but nevertheless entails the Einstein relation, a Maxwellian conditional steady state for the velocities, and the equipartition theorem. The over-damping limit leads to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Ito, Stratonovich or else). At odds with the latter interpretations, the corresponding Fokker-Planck equation admits an equilibrium steady state; a detailed comparison with other theories of state-dependent diffusion is carried out. We propose this as a theory of diffusion in a heat bath with varying temperature. Besides equilibrium, a crucial experimental signature is the non-uniform steady spatial distribution., Comment: 24 pages

We assume that markovian dynamics on a finite graph enjoys a gauge symmetry under local scalings of the probability density, derive the transformation law for the transition rates and interpret the thermodynamic force as a gauge potential. A widely accepted expression for the total entropy production of a system arises as the simplest gauge-invariant completion of the time derivative of Gibbs's entropy. We show that transition rates can be given a simple physical characterization in terms of locally-detailed-balanced heat reservoirs. It follows that Clausius's measure of irreversibility along a cyclic transformation is a geometric phase. In this picture, the gauge symmetry arises as the arbitrariness in the choice of a prior probability. Thermostatics depends on the information that is disposable to an observer; thermodynamics does not., Comment: 6 pages. Non-fatal errors in eq.(6), eq.(26) and eq.(31) have been amended

On networks representing probability currents between states of a system, we generalize Schnakenberg's theory of nonequilibrium observables to nonsteady states, with the introduction of a new set of macroscopic observables that, for planar graphs, are related by a duality. We apply this duality to the linear regime, obtaining a dual proposition for the minimum entropy production principle, and to discrete electromagnetism, finding that it exchanges fields with sources. We interpret duality as reversing the role of system and environment, and discuss generalization to nonplanar graphs. The results are based on two theorems regarding the representation of bilinear and quadratic forms over the edge vector space of an oriented graph in terms of observables associated to cycles and cocycles., Comment: 13 pages, 2 figures

In an essential and quite general setup, based on networks, we identify Schnakenberg's observables as the constraints that prevent a system from relaxing to equilibrium, showing that, in the linear regime, steady states satisfy a minimum entropy production principle. The result is applied to master equation systems, opening a new path to a well-known version of the principle regarding invariant states. Moreover, with the aid of a simple example, the principle is shown to conform to Prigogine's original formulation. Finally, we discuss analogies and differences with a recently proposed maximum entropy production principle., Comment: 10 pages, 1 figure. Published version

We generalize an orthonormality relation between decay eigenmodes of equilibrium systems to nonequilibrium markovian generators which commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical ensembles, we relate the result to the choice of a coordinate patch which makes the Fisher-Rao metric euclidean at the invariant state, realizing a sort of statistical equivalence principle. Finally, we classify nonequilibrium systems according to their spectrum, arguing that a degenerate Fisher matrix is a signature of the insurgence of a class of phase transitions between nonequilibrium regimes. We exhibit an order parameter with power-law critical decay and prove divergent correlations between suitable unbiased estimators., Comment: This paper has been withdrawn by the author. New vastly revised version of this paper: Fisher information of Markovian decay modes - Nonequilibrium equivalence principle, dynamical phase transitions and coarse graining, arXiv:1409.4193