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.

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 among others the determination of the conservation laws for the serial and parallel associations of thermodynamic devices.

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 pseudolinear subnetworks, called chemical modules, in two different ways: First by computing the nonequilibrium conductances of the modules that are then serially connected and second by computing the nonequilibrium conductance of the CRN directly. The two approaches coincide, as expected from our theory of thermodynamic circuits.

Starting from Ioffe’s description of a thermoelectric converter, we recover the optimal working points of conversion: the point of maximum efficiency and the one of maximal power. Inspired by biological converters’ optimization, we compute a third optimal point associated with cost of energy (COE). This alternative cost function corresponds to the amount of heat exchanged with the cold reservoir per unit of electric current used. This work emphasizes the symmetry between the efficiency and performance coefficient of the electric generator and heat pump modes. It also reveals the relation between their optimal working points.

Following up on the recently published circuit theory for thermodynamic devices, we consider networks of thermoelectric converters (TECs) in stationary nonequilibrium. Assuming constant thermoelectric properties, the integration over a finite thickness of the linear local response of the thermoelectric material yields the nonlinear current-force characteristics. We show how to derive a choice of nonequilibrium conductance matrix summarizing the current-force characteristics for all 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 (parallel) association of two TECs using the laws of resistance (conductance) matrix addition. For TECs in series, we find current-dependent boundary conditions for each subdevice. 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.

Maxwell’s demon is a famous thought experiment and a paradigm of the thermodynamics of information. It is related to Szilard’s engine, a two-state information-to-work conversion device in which the demon performs single measurements and extracts work depending on the state measurement outcome. A variant of these models, the continuous Maxwell demon (CMD), was recently introduced by Ribezzi-Crivellari and Ritort where work was extracted after multiple repeated measurements every time that τ is in a two-state system. The CMD was able to extract unbounded amounts of work at the cost of an unbounded amount of information storage. In this work, we built a generalization of the CMD to the N-state case. We obtained generalized analytical expressions for the average work extracted and the information content. We show that the second law inequality for information-to-work conversion is fulfilled. We illustrate the results for N-states with uniform transition rates and for the N = 3 case.