On the (Boltzmann) entropy of non-equilibrium systems

S Goldstein; Joel L Lebowitz

doi:10.1016/j.physd.2004.01.008

Distinguishing between Clausius, Boltzmann and Pauling Entropies of Frozen Non-Equilibrium States

Entropy ◽

10.3390/e21080799 ◽

2019 ◽

Vol 21 (8) ◽

pp. 799 ◽

Cited By ~ 1

Author(s):

Rainer Feistel

Keyword(s):

Heat Capacity ◽

Historical Review ◽

Zero Point ◽

Quantum States ◽

Boltzmann Entropy ◽

Nernst Theorem ◽

Symbolic Information ◽

Equilibrium Structures ◽

Physical Quantities ◽

Non Equilibrium

In conventional textbook thermodynamics, entropy is a quantity that may be calculated by different methods, for example experimentally from heat capacities (following Clausius) or statistically from numbers of microscopic quantum states (following Boltzmann and Planck). It had turned out that these methods do not necessarily provide mutually consistent results, and for equilibrium systems their difference was explained by introducing a residual zero-point entropy (following Pauling), apparently violating the Nernst theorem. At finite temperatures, associated statistical entropies which count microstates that do not contribute to a body’s heat capacity, differ systematically from Clausius entropy, and are of particular relevance as measures for metastable, frozen-in non-equilibrium structures and for symbolic information processing (following Shannon). In this paper, it is suggested to consider Clausius, Boltzmann, Pauling and Shannon entropies as distinct, though related, physical quantities with different key properties, in order to avoid confusion by loosely speaking about just “entropy” while actually referring to different kinds of it. For instance, zero-point entropy exclusively belongs to Boltzmann rather than Clausius entropy, while the Nernst theorem holds rigorously for Clausius rather than Boltzmann entropy. The discussion of those terms is underpinned by a brief historical review of the emergence of corresponding fundamental thermodynamic concepts.

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Application of analytical electron microscopy to studies of equilibrium and non-equilibrium segregation in materials

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100130705 ◽

1992 ◽

Vol 50 (2) ◽

pp. 1214-1215

Author(s):

Edward A Kenik

Keyword(s):

Electron Microscopy ◽

Analytical Electron Microscopy ◽

Extended Defects ◽

Probe Diameter ◽

Specimen Holder ◽

Analytical Electron ◽

Scanning Transmission ◽

Solute Atoms ◽

Equilibrium Segregation ◽

Non Equilibrium

Segregation of solute atoms to grain boundaries, dislocations, and other extended defects can occur under thermal equilibrium or non-equilibrium conditions, such as quenching, irradiation, or precipitation. Generally, equilibrium segregation is narrow (near monolayer coverage at planar defects), whereas non-equilibrium segregation exhibits profiles of larger spatial extent, associated with diffusion of point defects or solute atoms. Analytical electron microscopy provides tools both to measure the segregation and to characterize the defect at which the segregation occurs. This is especially true of instruments that can achieve fine (<2 nm width), high current probes and as such, provide high spatial resolution analysis and characterization capability. Analysis was performed in a Philips EM400T/FEG operated in the scanning transmission mode with a probe diameter of <2 nm (FWTM). The instrument is equipped with EDAX 9100/70 energy dispersive X-ray spectrometry (EDXS) and Gatan 666 parallel detection electron energy loss spectrometry (PEELS) systems. A double-tilt, liquid-nitrogen-cooled specimen holder was employed for microanalysis in order to minimize contamination under the focussed spot.

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