We study the thermodynamic properties induced by non-reciprocal interactions between stochastic degrees of freedom in time- and space-continuous systems. We show that, under fairly general conditions, non-reciprocal coupling alone implies a steady energy flow through the system, i.e., non-equilibrium. Projecting out the non-reciprocally coupled degrees of freedom renders non-Markovian, one-variable Langevin descriptions with complex types of memory, for which we find a generalized second law involving information flow. We demonstrate that non-reciprocal linear interactions can be used to engineer non-monotonic memory, which is typical for, e.g., time-delayed feedback control, and is automatically accompanied with a nonzero information flow through the system. Furthermore, already a single non-reciprocally coupled degree of freedom can extract energy from a single heat bath (at isothermal conditions), and can thus be viewed as a minimal version of a time-continuous, autonomous "Maxwell demon". At the same time, the non-reciprocal system has characteristic features of active matter, such as a positive energy input on the level of the flucuating trajectories, without global particle transport.
Stochastic line integrals provide a useful tool for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here, we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of steady-state probability c…
Non-reciprocal interactions are present in many systems out of equilibrium. The rate of entropy production is a measure that quantifies the time irreversibility of a system, and thus how far it is from equilibrium. In this work, we introduce a non-motile active particle system where activity originates from asymmetric, pairwise interaction forces that result in an injection of energy at the microscopic scale. We calculate stationary correlation functions and entropy production rate in three exactly solvable cases, and analyse a more general case in a perturbation theory as an expansion in weak…
In classical thermodynamic processes the unavoidable presence of irreversibility, quantified by the entropy production, carries two energetic footprints: the reduction of extractable work from the optimal, reversible case, and the generation of a surplus of heat that is irreversibly dissipated to the environment. Recently it has been shown that in the quantum regime an additional quantum irreversibility occurs that is linked to decoherence into the energy basis. Here we employ quantum trajectories to construct distributions for classical heat and quantum heat exchanges, and show that the heat …
We show that memory, feedback, and activity are all describable by the same unifying concept, that is non-reciprocal (NR) coupling. We demonstrate that characteristic thermodynamic features of these intrinsically nonequilibrium systems are reproduced by low-dimensional Markovian networks with NR coupling, which we establish as minimal models for such complex systems. NR coupling alone implies a violation of the fluctuation-dissipation relation, which is inevitably connected to entropy production, i.e., irreversibility. Hiding the NR coupled degrees of freedom renders non-Markovian, one-variabl…
In this paper, the emergence of macroscopic-scale pseudo time-irreversibility is studied in the closed classical many-body system of pair interacting particles. First, exact continuum equations are derived to the Hamiltonian dynamics without the utilisation of statistical mechanics. Next, it is shown that the momentum density field incorporates the thermal degrees of freedom in the considered scaling limit, and therefore initial condition indicated pseudo time-irreversible solutions may exist to the dynamical equations. Numerical evidence for pseudo-irreversible thermal equilibration, heat and…
We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate $q$, momentum $p$, and thermostat control variable $ζ$. The second type simulates a conservative two-dimensional $N$-body fluid with $4N$ phase variables $\{q,p\}$ undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the $4N$ manybody motion equations both types of simulation are irreversible, obeying th…
We discuss the non-equilibrium properties of a thermally driven micromachine consisting of three spheres which are in equilibrium with independent heat baths characterized by different temperatures. Within the framework of a linear stochastic Langevin description, we calculate the time-dependent average irreversibility that takes a maximum value for a finite time. This time scale is roughly set by the spring relaxation time. The steady-state average entropy production rate is obtained in terms of the temperatures and the friction coefficients of the spheres. The average entropy production rate…
Developments in the thermodynamics of small quantum systems envisage non-classical thermal machines. In this scenario, energy fluctuations play a relevant role in the description of irreversibility. We experimentally implement a quantum heat engine based on a spin-1/2 system and nuclear magnetic resonance techniques. Irreversibility at microscope scale is fully characterized by the assessment of energy fluctuations associated with the work and heat flows. We also investigate the efficiency lag related to the entropy production at finite time. The implemented heat engine operates in a regime wh…
Microscopic Dynamical Entropy I: Quantifying Hamiltonian Irreversibility in Large and Small Systems
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；DAMTP, Centre for Mathematical Sciences, University of Cambridge
Irreversibility is a critical property of non-equilibrium transport processes. An opinion has long been insisted that the entropy production rate is a Lyapunov function for all kinds of processes, that is, the principle of minimum entropy production. However, such principle is based on some strong assumptions that are rarely valid in practice. Here, the common features of parabolic-like transport processes are discussed. A theorem is then put forward that the dot products of fluxes and corresponding forces serve as Lyapunov function for parabolic-like transport processes. Such fluxes and force…
The hallmark of active matter is the autonomous directed motion of its microscopic constituents driven by consumption of energy resources. This leads to the emergence of large scale dynamics and structures without any equilibrium equivalent. Though active field theories offer a useful hydrodynamic description, it is unclear how to properly quantify the energetic cost of the dynamics from such a coarse-grained description. We provide a thermodynamically consistent framework to identify the energy exchanges between active systems and their surrounding thermostat at the hydrodynamic level. Based …
The application of a temperature gradient along a fluid-solid interface generates stresses in the fluid causing "thermo-osmotic" flow. Much of the understanding of this phenomenon is based on Derjaguin's work relating thermo-osmotic flows to the mechano-caloric effect, namely, the interfacial heat flow induced by a pressure gradient. This is done by using Onsager's reciprocity relationship for the equivalence of the thermo-osmotic and mechano-caloric cross-term transport coefficients. Both Derjaguin theory and Onsager framework for out-of-equilibrium systems are formulated in macroscopic therm…
Even though irreversibility is one of the major hallmarks of any real life process, an actual understanding of irreversible processes remains still mostly semiempirical. In this paper we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the time needed to run through such an irreversible cycle multiplied by the irreversible work lost in the cycle, is bounded from below by an irreducible and process-dependent constant that has the dimension of an action. The constant in question depends on a…
We demonstrate that the reformulation of renormalization group (RG) flow equations as non-linear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional $\mathbb{Z}_2$-symmetric model that the dissipative character of generic non-linear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semi-group property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dis…
Information dynamics is an emerging description of information processing in complex systems which describes systems in terms of intrinsic computation, identifying computational primitives of information storage and transfer. In this paper we make a formal analogy between information dynamics and stochastic thermodynamics which describes the thermal behaviour of small irreversible systems. As stochastic dynamics is increasingly being utilized to quantify the thermodynamics associated with the processing of information we suggest such an analogy is instructive, highlighting that existing thermo…
