Niet-Evenwichts Grafenftheorie en Derde Wet van de Thermodynamica

This thesis delves into the study of thermal response of (classical) steady nonequilibrium systems at low temperatures. Our analysis focuses on Markov jump processes, which are relevant as models of low-temperature open-driven systems in the incoherent approximation of quantum systems. A weighted directed graph can be assigned to such a Markov jump process. That leads to graphical representations of stationary distributions, of steady currents and of the excess heat. More specifically, we give graphical representations of the solution of the Poisson equation (for the Markov generator). In that way, we manage to derive (very) low-temperature properties of the heat capacity for a relevant class of driven and active systems: we could derive a nonequilibrium (extension of the) Third Law of Thermodynamics, motivated by and similar to the Nernst heat theorem in thermal equilibrium.<br /> After the general introduction, also containing the outline of the thesis, we start with a more formal chapter on Markov jump processes to describe the setup of the thesis. That leads us into the world of nonequilibrium graph theory. We rewrite the Kirchhoff formula for stationary distributions in terms of a tree ensemble. We derive a McLennan-tree characterization for close-to-equilibrium processes. We obtain graphical response formulae for the stationary distribution in the asymptotic regime of large driving. Furthermore, graphical expressions of currents and traffic are given, allowing us to study them in (other) asymptotic regimes. We continue in nonequilibrium graph theory with expressions for the solution of the Poisson equation for Markov jump processes. More precisely, for a given centered f (for which the stationary mean equals zero), we provide a graphical representation of the solution of the Poisson equation LV + f=0, where L is the backward generator of an irreducible (finite) Markov jump process. We call V the quasipotential in case $f$ represents an excess heat, which is the crucial quantity for our extended Third Law. In addition, we derive a graphical representation of the mean first-passage time, leading to estimates on time-accumulated differences during relaxation toward a steady nonequilibrium regime. That also gives bounds on the quasipotential, which is central in our derivation of the extended Third Law. Next, we establish our extended Third Law, characterizing the low-temperature thermal response of a nonequilibrium system (as characterized by a Markov jump process). Using the local detailed balance condition, the heat flux is identified. We then introduce the steady nonequilibrium heat capacity, which prompts us to investigate its properties at very low temperatures. The extended Third Law has two conditions under which the steady nonequilibrium heat capacity vanishes at zero absolute temperature. In other words, we discuss via general arguments and examples when and why the steady nonequilibrium heat capacity vanishes with temperature. Having a counter-example for the Third Law is already natural in equilibrium, where e.g. spin ice does not satisfy it. Similar to the equilibrium version, the dominant stationary distribution close to zero ambient temperature must be (sufficiently) nondegenerate. Yet, in nonequilibrium, we have an extra and dynamic condition: the dynamical activity and accessibility of the dominant state at low temperatures must be maintained sufficiently high so that relaxation times do not start to differ dramatically between different initial states. We must ensure that relaxation times do not exceed the dissipation time. In continuation, we study the heat capacity for close-to-equilibrium systems, i.e., in the so-called McLennan regime. It will be proved that in this regime, the heat capacity at low temperatures always satisfies the second condition, meaning that the violation of the extended Third Law is indeed a nonlinear effect in the nonequilibrium driving. The mathematical proof of the extended Third Law in the quasistatic limit comes later. As mentioned above, the Poisson equation is solved for specific f, related to the dissipated power, and the solution is the quasipotential V. The graphical representation of the quasipotential split into tree-terms and loop-terms. This new representation is used to establish the present second condition of the extended Third Law.