For a two-level quantum mechanical system, we derive microscopically the exact expression for the fluctuation of microscopic work in a multi-step non-equilibrium process, and we rigorously prove that in an isothermal process, the fluctuation is vanishingly small, and the most probabilistic work just equals to the difference of the free energy. Our study demonstrates that the convergence of the microscopic work in the isothermal process is due to the nature of isothermal process rather than usual thermodynamic limit condition. Our investigation justifies the validity of "minimum work principle" formulation of the second law even for a small system far from thermodynamic limit.
We obtain macroscopic isothermal thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics in contact with a heat bath. The microscopic dynamics is given by a chain of anharmonic oscillators subject to a varying tension (external force) and the contact with the heat bath is modeled by independent Langevin dynamics acting on each particle. After a diffusive space-time scaling and cross-graining, the profile of volume converges to the solution of a deterministic diffusive equation with boundary conditions given by the applied tension. This defines an irreversible…
We consider a chain of anharmonic oscillators immersed in a heat bath with a temperature gradient and a time varying tension applied to one end of the chain while the other side is fixed to a point. We prove that under diffusive space-time rescaling the volume strain distribution of the chain evolves following a non-linear diffusive equation. The stationary states of the dynamics are of non-equilibrium and have a positive entropy production, so the classical relative entropy methods cannot be used. We develop new estimates based on entropic hypocoercivity, that allows to control the distributi…
We derive a microscopic expression of the Clausius inequality for isothermal open systems within the fundamental Hamiltonian dynamics. We also show the positivity of the dissipative work, which is a path integral generalization of the so-called relative entropy for the probability functionals of time forward- and reversed- trajectories.
For an understanding of a heat engine working in the microscopic scale, it is often necessary to estimate the amount of reversible work extracted by isothermal expansion of the quantum gas used as its working substance. We consider an engine with a movable wall, modeled as an infinite square well with a delta peak inside. By solving the resulting one-dimensional Schrödinger equation, we obtain the energy levels and the thermodynamic potentials. Our result shows how quantum tunneling degrades the engine by decreasing the amount of reversible work during the isothermal expansion.
We study microscopic engines that use a single active particle as their "working medium". Part of the energy required to drive the directed motion of the particle can be recovered as work, even at constant temperature. A wide class of synthetic active particles can be captured by schematically accounting for the chemical degrees of freedom that power the directed motion without having to resolve the exact microscopic mechanism. We derive analytical results for the quasi-static thermodynamic efficiency, i.e., the fraction of available chemical energy that can be recovered as mechanical work. Wh…
Stochastic thermodynamics extends the notions and relations of classical thermodynamics to small systems that experience strong fluctuations. The definitions of work and heat and the microscopically reversible condition are two key concepts in the current framework of stochastic thermodynamics. Herein, we apply stochastic thermodynamics to small systems with odd controlling parameters and find that the definition of heat and the microscopically reversible condition are incompatible. Such a contradiction also leads to a revision to the fluctuation theorems and nonequilibrium work relations. By …
The isothermal compressibility of a general crystal is analyzed within classical density functional theory. Our approach can be used for homogeneous and unstrained crystals containing an arbitrarily high density of local defects. We start by coarse-graining the microscopic particle density and then obtain the long wavelength limits of the correlation functions of elasticity theory and the thermodynamic derivatives. We explicitly show that the long wavelength limit of the microscopic density correlation function differs from the isothermal compressibility. It also cannot be obtained from the st…
We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. Starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the def…
The isothermal compressibility of a general crystal is derived and discussed within (classical) density functional theory. Starting from the microscopic particle density, we carefully coarse grain to obtain the thermodynamic compressibility and the long wavelength limits of the correlation functions of elasticity theory. We explicitly show that the isothermal compressibility is not the wavevector to zero limit of the (total) density correlation function. The latter also cannot be obtained from the static structure factor measured in a scattering experiment.
Carnot's four-part ideal-gas cycle includes both isothermal and adiabatic expansions and compressions. Analyzing this cycle provides the fundamental basis for statistical thermodynamics. We explore the cycle here from a pedagogical view in order to promote understanding of the macroscopic thermodynamic entropy, the state function associated with thermal energy changes. From the alternative microscopic viewpoint the Hamiltonian ${\cal H}(q,p)$ is the energy and entropy is the (logarithm of the) phase-space volume $Ω$ associated with a macroscopic state. We apply two novel forms of Hamiltonian m…
We revisit the paradigm of an ideal gas under isothermal conditions. A moving piston performs work on an ideal gas in a container that is strongly coupled to a heat reservoir. The thermal coupling is modelled by stochastic scattering at the boundaries. In contrast to recent studies of an adiabatic ideal gas with a piston [R.C. Lua and A.Y. Grosberg, \textit{J. Phys. Chem. B} 109, 6805 (2005); I. Bena et al., \textit{Europhys. Lett.} 71, 879 (2005)], container and piston stay in contact with the heat bath during the work process. Under this condition the heat reservoir as well as the system dep…
We present a novel picture of a non isothermal solidification process starting from a molecular level, where the microscopic origin of the basic mechanisms and of the instabilities characterizing the approach to equilibrium is rendered more apparent than in existing approaches based on coarse grained free energy functionals à la Landau. The system is composed by a lattice of Potts spins, which change their state according to the stochastic dynamics proposed some time ago by Creutz. Such a method is extended to include the presence of latent heat and thermal conduction. Not only the model agree…
We determine the statistics of work in isothermal volume changes of a classical ideal gas consisting of a single particle. Combining our results with the findings of Lua and Grosberg [J. Chem. Phys. B 109, 6805 (2005)] on adiabatic expansions and compressions we then analyze the joint probability distribution of heat and work for a microscopic, non-equilibrium Carnot cycle and determine its efficiency at maximum power.
Diffusive motion in an externally driven potential is considered. It is shown that the distribution of work required to drive the system from an initial equilibrium state to another is Gaussian for slow but finite driving. Our result is obtained by projection method techniques exploiting a small parameter defined as the switching rate between the two states of the system. The exact solution for a simple model system shows that such an expansion may fail in higher orders, since the mean and the variance following from the exact distribution show non-analytic behavior.
We investigate the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large …
We consider an isothermal machine composed of two Brownian particles (say particle A and B) connected by a harmonic spring. A constant load is attached to particle A, and the particle B is trapped in a harmonic confinement whose minimum is dragged with a constant velocity. Whole system is in contact with the heat bath of a constant temperature. We obtain the distribution of the work done on particle A and particle B, and transient fluctuation theorem for these quantities is tested in the weak coupling limit and for both small and large observation time. Moreover, we show that the transient flu…
Besides the growing interest in old concepts like temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. With the electronic circuits and computer chips getting smaller and smaller, a fresh look should be appropriate on the equilibrium and nonequilibrium thermodynamics at small length scales far below the thermodynamic limit, i.e. on the theoretical understanding of original macroscopic processes, e.g. transport of energy, heat, charge, mass, magnetization etc. Only from the foundations of a theory its limits of applicability m…
The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied system is coupled to a two-level system (probe) as a microscopic thermometer. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. This temperat…
The distribution function P(j) of the microscopic energy flux, j, in equilibrium state is studied. It is observed that P(j) has a broad peak in small j regime and a stretched-exponential decay for large j. The peak structure originates in a potential advection term and energy transfer term between the particles. The stretched exponential tail comes from the momentum energy advection term.
