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Entropy

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...

Entropy

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Maxwell's thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell's relations relates a partial differential between quantities that can be hard to measure experimentally to a partial differential between quantities that can be easily measured. These relations are a set of equations derivable from the symmetry of the second derivatives and the thermodynamic potentials.
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Related Experiment Video

Updated: May 23, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Modified fluctuation-dissipation and Einstein relation at nonequilibrium steady states.

Debasish Chaudhuri¹, Abhishek Chaudhuri

¹FOM Institute for Atomic and Molecular Physics, Science Park 104, NL-1098XG Amsterdam, The Netherlands. d.chaudhuri@amolf.nl

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|April 3, 2012

Summary

This summary is machine-generated.

This study unifies modified fluctuation-dissipation relations (MFDR) for nonequilibrium systems. It demonstrates the equivalence of different dynamic approaches and corrects the Einstein relation using a flashing ratchet model.

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Area of Science:

Statistical Mechanics
Nonlinear Dynamics
Physical Chemistry

Background:

The fluctuation-dissipation relation (FDR) traditionally connects equilibrium fluctuations to response functions.
Extending FDR to nonequilibrium steady states (NESS) is crucial for understanding complex systems.
Previous work has explored modified FDR (MFDR) but lacked a unified derivation across different dynamics.

Purpose of the Study:

To present a unified theoretical framework for deriving modified fluctuation-dissipation relations (MFDR).
To demonstrate the equivalence of MFDR derived from continuum Langevin and discrete master equation dynamics.
To provide a corrected Einstein relation applicable to nonequilibrium systems.

Main Methods:

Unified derivation of MFDR based on pioneering work by Agarwal.
Comparison of velocity forms of MFDR obtained from Langevin and master equation approaches.
Application of the derived formalism to a flashing ratchet model.

Main Results:

A unified derivation of several modified fluctuation-dissipation relations (MFDR) is presented.
Equivalence between velocity forms of MFDR derived from continuum Langevin and discrete master equation dynamics is shown.
An additive correction to the Einstein relation is derived and exemplified.

Conclusions:

The developed formalism provides a consistent approach to MFDR for nonequilibrium systems.
The equivalence of different dynamical descriptions simplifies the analysis of NESS.
The corrected Einstein relation offers a valuable tool for studying molecular motors and similar systems.