Matteo Polettini

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 or pump), the most probable efficiency decreasing in time. The limit ζ→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 the Carnot limit at finite entropy production.

Nonequilibrium thermodynamics as a gauge theory

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 Gibbss entropy. We show that transition rates can be given a simple physical characterization in terms of locally detailed balanced heat reservoirs. It follows that Clausiuss 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.

Fact-Checking Ziegler’s Maximum Entropy Production Principle beyond the Linear Regime and towards Steady States

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.

Generally covariant state-dependent diffusion

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 external 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 incorporates the Einstein relation, a Maxwellian conditional steady state for the velocities, and the equipartition theorem. The overdamping limit leads to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Itō, Stratonovich or other). 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 nonuniform steady spatial distribution.

Nonconvexity of the relative entropy for Markov dynamics: a Fisher information approach.

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 disprove the hypotheses that a general evolution principle of thermodynamics based on the decrease of the nonadiabatic entropy production could hold. 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 admit nonconvex solutions and those that do not. 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 satisfying 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 by Maes et al. [Phys. Rev. Lett. 107, 010601 (2011)] regarding monotonicity of the large deviation rate functional for the occupation probability, showing that while the conjecture might hold in the long-time limit, the conditions for Lyapunovs second criterion for stability are not met.

Tightening the uncertainty principle for stochastic currents

We connect two recent advances in the stochastic analysis of nonequilibrium systems: the (loose) uncertainty principle for the currents, which states that statistical errors are bounded by thermodynamic dissipation, and the analysis of thermodynamic consistency of the currents in the light of symmetries. Employing the large deviation techniques presented by Gingrich et al. [Phys. Rev. Lett. 116, 120601 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.120601] and Pietzonka, Barato, and Seifert [Phys. Rev. E 93, 052145 (2016)2470-004510.1103/PhysRevE.93.052145], we provide a short proof of the loose uncertainty principle, and prove a tighter uncertainty relation for a class of thermodynamically consistent currents J. Our bound involves a measure of partial entropy production, that we interpret as the least amount of entropy that a system sustaining current J can possibly produce, at a given steady state. We provide a complete mathematical discussion of quadratic bounds which allows one to determine which are optimal, and finally we argue that the relationship for the Fano factor of the entropy production rate varσ/meanσ≥2 is the most significant realization of the loose bound. We base our analysis both on the formalism of diffusions, and of Markov jump processes in the light of Schnakenbergs cycle analysis.

Macroscopic constraints for the minimum entropy production principle.

In an essential and quite general setup, based on networks, we identify Schnakenbergs 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 Prigogines original formulation. Finally, we discuss analogies and differences with a recently proposed maximum entropy production principle.

Cycle/Cocycle Oblique Projections on Oriented Graphs

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.

Conservation laws and symmetries in stochastic thermodynamics

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 systems 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 provides 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.

Carnot efficiency at divergent power output

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.