Thermodynamics Far from the Thermodynamic Limit

Abstract

Understanding how small systems exchange energy with a heat bath is important to describe how their unique properties can be affected by the environment. In this contribution, we apply Landsberg's theory of temperature-dependent energy levels to describe the progressive thermalization of small systems as their spectrum is perturbed by a heat bath. We propose a mechanism whereby the small system undergoes a discrete series of excitations and isentropic spectrum adjustments leading to a final state of thermal equilibrium. This produces standard thermodynamic results without invoking system size. The thermal relaxation of a single harmonic oscillator is analyzed as a model example of a system with a quantized spectrum than can be embedded in a thermal environment. A description of how the thermal environment affects the spectrum of a small system can be the first step in using environmental factors, such as temperature, as parameters in the design and operation of nanosystem properties.2

Internal geometric friction in a Kitaev-chain heat engine

“…The method of TDELs was proposed as a fast and convenient way to perform statistical mechanic calculations for an assembly of systems [22,23]. It was applied to semiconductors [33][34][35][36], superfluids [37], optomechanical oscillators [38,39], heat losses in thermoelectric systems [27,28], and thermalization of finite-size systems [24][25][26]. Even before TDELs were proposed, the advantages of temperature-dependent mean potentials and corresponding forces were known in statistical mechanics of fluid mixtures [40].…”

“…Accordingly, the second term cannot be interpreted as heat; it represents the work applied to change the energy gaps. Such a mechanism has been discussed in detail to explain thermalization of small systems [24][25][26] Equation ( 23) originates from the first law modified for a small system; rewriting it as dS T = dQ/T − ∂H/∂T dT /T , one recovers the modified second law of information transfer channels [29]. The entropy transferred from a heat source into the system through the boundary is balanced by the entropy lost at the boundary.…”

“…The connection between Hill's nanothermodynamics and TDELs is sustained by recognizing that the subdivision potential acts as a thermal perturbation by the environment [24][25][26]. As explained in Sec.…”

“…Furthermore, we address the influence of a possible interface that connects the heat baths to the Kitaev chain on the heat transfer. The later question is investigated by another finite-size thermodynamics method, known as temperaturedependent energy levels (TDELs) [22][23][24][25][26]. The TDELs scheme takes into account the contribution of the bath-system interface as another energy dissipation channel [27,28], consistent with the extended second law of thermodynamics [29].…”

Heat capacities of thermally manipulated mechanical oscillator at strong coupling

“…At the same time, thermodynamics of systems at strong coupling has recently attracted a significant attention and a number of general theoretical studies has appeared, [12][13][14][15][16][17][18][19][20] which generalized classical theoretical works, [21][22][23][24][25][26][27] and several particular models with temperature-dependent energy levels have been explored. [28][29][30][31][32] However, the fundamental theoretical approaches encounter severe difficulties already at the level of definitions of basic thermodynamic quantities, 20 where the concepts of heat and entropy production cannot be unambiguously identified. A promising way out of these theoretical struggles is, in our opinion, to propose and build in lab real-world quantum-mechanical thermodynamic devices.…”