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Local non-equilibrium thermodynamics.

Jinwoo L, Tanaka H - Sci Rep (2015)

Bottom Line: However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally.Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials.We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Kwangwoon University, 20 Kwangwoon-ro, Nowon-gu, Seoul 139-701, Korea.

ABSTRACT
Local Shannon entropy lies at the heart of modern thermodynamics, with much discussion of trajectory-dependent entropy production. When taken at both boundaries of a process in phase space, it reproduces the second law of thermodynamics over a finite time interval for small scale systems. However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally. Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

No MeSH data available.


A universal frame of thermodynamics and a fundamental question.Thermodynamics is a theory for ensembles that are composed of a large number of fluctuating degrees of freedom. An ensemble is constrained by a macrostate λ. In equilibrium thermodynamics, λ varies in a quasi-static manner. In the modern context, the theory considers arbitrary time-varying processes. Two typical systems in both approaches are shown schematically. On the left, gas particles are fluctuating in a cylinder. The macrostate may be specified by the temperature, the volume of the cylinder, and the number of particles. On the right, a polymer chain is fluctuating under the influence of optically controlled bead connected to the end of the chain. The macrostate may be specified by the temperature, and the location of the bead applying force to the molecule. A fundamental question that we investigate is how we can justify thermodynamic descriptions of fluctuating microstates.
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f1: A universal frame of thermodynamics and a fundamental question.Thermodynamics is a theory for ensembles that are composed of a large number of fluctuating degrees of freedom. An ensemble is constrained by a macrostate λ. In equilibrium thermodynamics, λ varies in a quasi-static manner. In the modern context, the theory considers arbitrary time-varying processes. Two typical systems in both approaches are shown schematically. On the left, gas particles are fluctuating in a cylinder. The macrostate may be specified by the temperature, the volume of the cylinder, and the number of particles. On the right, a polymer chain is fluctuating under the influence of optically controlled bead connected to the end of the chain. The macrostate may be specified by the temperature, and the location of the bead applying force to the molecule. A fundamental question that we investigate is how we can justify thermodynamic descriptions of fluctuating microstates.

Mentions: Statistical mechanics provides physical interpretations of entropy and free energy that are macro-state functions (i.e., functions defined on a domain of the phase-space points of a system, and thus inevitably non-local in character), and sets bounds on permissible processes expressed as path functions like heat and work, where a path is defined as a trajectory of macrostates of a system1. As a system gets smaller, the effect of fluctuations becomes significant, yet modern theory234 provides permissible distributions of fluctuating path functions in the form of beautiful equalities5678910. Modern theory has identified energetics11 and entropy production on the level of individual trajectories12, and has linked path functions to properties of macrostates13, as verified experimentally141516171819. The relationship between classical and modern approaches is schematically drawn in Fig. 1.


Local non-equilibrium thermodynamics.

Jinwoo L, Tanaka H - Sci Rep (2015)

A universal frame of thermodynamics and a fundamental question.Thermodynamics is a theory for ensembles that are composed of a large number of fluctuating degrees of freedom. An ensemble is constrained by a macrostate λ. In equilibrium thermodynamics, λ varies in a quasi-static manner. In the modern context, the theory considers arbitrary time-varying processes. Two typical systems in both approaches are shown schematically. On the left, gas particles are fluctuating in a cylinder. The macrostate may be specified by the temperature, the volume of the cylinder, and the number of particles. On the right, a polymer chain is fluctuating under the influence of optically controlled bead connected to the end of the chain. The macrostate may be specified by the temperature, and the location of the bead applying force to the molecule. A fundamental question that we investigate is how we can justify thermodynamic descriptions of fluctuating microstates.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4296294&req=5

f1: A universal frame of thermodynamics and a fundamental question.Thermodynamics is a theory for ensembles that are composed of a large number of fluctuating degrees of freedom. An ensemble is constrained by a macrostate λ. In equilibrium thermodynamics, λ varies in a quasi-static manner. In the modern context, the theory considers arbitrary time-varying processes. Two typical systems in both approaches are shown schematically. On the left, gas particles are fluctuating in a cylinder. The macrostate may be specified by the temperature, the volume of the cylinder, and the number of particles. On the right, a polymer chain is fluctuating under the influence of optically controlled bead connected to the end of the chain. The macrostate may be specified by the temperature, and the location of the bead applying force to the molecule. A fundamental question that we investigate is how we can justify thermodynamic descriptions of fluctuating microstates.
Mentions: Statistical mechanics provides physical interpretations of entropy and free energy that are macro-state functions (i.e., functions defined on a domain of the phase-space points of a system, and thus inevitably non-local in character), and sets bounds on permissible processes expressed as path functions like heat and work, where a path is defined as a trajectory of macrostates of a system1. As a system gets smaller, the effect of fluctuations becomes significant, yet modern theory234 provides permissible distributions of fluctuating path functions in the form of beautiful equalities5678910. Modern theory has identified energetics11 and entropy production on the level of individual trajectories12, and has linked path functions to properties of macrostates13, as verified experimentally141516171819. The relationship between classical and modern approaches is schematically drawn in Fig. 1.

Bottom Line: However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally.Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials.We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Kwangwoon University, 20 Kwangwoon-ro, Nowon-gu, Seoul 139-701, Korea.

ABSTRACT
Local Shannon entropy lies at the heart of modern thermodynamics, with much discussion of trajectory-dependent entropy production. When taken at both boundaries of a process in phase space, it reproduces the second law of thermodynamics over a finite time interval for small scale systems. However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally. Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

No MeSH data available.