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A No-Go Theorem of Analytical Mechanics for the Second Law Violation

P. D. Gujrati

Abstract

We follow the Boltzmann-Clausius-Maxwell (BCM) proposal to solve a long-standing problem of identifying the underlying cause of the second law (SL) of spontaneous irreversibility, a stochastic universal principle, as the mechanical equilibrium (stable or unstable) principle (Mec-EQ-P) of analytical mechanics of an isolated nonequilibrium system of any size. The principle leads to nonnegative system intrinsic (SI) microwork and SI-average macrowork dW during any spontaneous process. In conjuction with the first law, Mec-EQ-P leads to a generalized second law (GSL) dQ=dW>0, where dQ=TdS is the purely stochastic SI-macroheat that corresponds to dS>0 for T>0 and dS<0 for T<0, where T is the temperature. The GSL supercedes the conventional SL formulation that is valid only for a macroscopic system for positive temperatures temperatures, but reformulates it to dS<0 for negative temperatures. It is quite surprising that GSL is not only a direct consequence of intertwined mechanical and stochastic macroquantities through the first law but also remains valid for any arbitrary irreversible process in a system of any size as an identity for positive and negative temperatures. It also becomes a no-go theorem for GSL-violation unless we abandon Mec-EQ-P of analytical mechanics used in the BCM proposal, which will be catastrophic for theoretical physics. In addition, Mec-EQ-P also provides new insights into the roles of spontaneity, nonspontaneity, negative temperatures, instability, and the significance of dS<0 due to nonspontaneity and inserting internal constraints.

A No-Go Theorem of Analytical Mechanics for the Second Law Violation

Abstract

We follow the Boltzmann-Clausius-Maxwell (BCM) proposal to solve a long-standing problem of identifying the underlying cause of the second law (SL) of spontaneous irreversibility, a stochastic universal principle, as the mechanical equilibrium (stable or unstable) principle (Mec-EQ-P) of analytical mechanics of an isolated nonequilibrium system of any size. The principle leads to nonnegative system intrinsic (SI) microwork and SI-average macrowork dW during any spontaneous process. In conjuction with the first law, Mec-EQ-P leads to a generalized second law (GSL) dQ=dW>0, where dQ=TdS is the purely stochastic SI-macroheat that corresponds to dS>0 for T>0 and dS<0 for T<0, where T is the temperature. The GSL supercedes the conventional SL formulation that is valid only for a macroscopic system for positive temperatures temperatures, but reformulates it to dS<0 for negative temperatures. It is quite surprising that GSL is not only a direct consequence of intertwined mechanical and stochastic macroquantities through the first law but also remains valid for any arbitrary irreversible process in a system of any size as an identity for positive and negative temperatures. It also becomes a no-go theorem for GSL-violation unless we abandon Mec-EQ-P of analytical mechanics used in the BCM proposal, which will be catastrophic for theoretical physics. In addition, Mec-EQ-P also provides new insights into the roles of spontaneity, nonspontaneity, negative temperatures, instability, and the significance of dS<0 due to nonspontaneity and inserting internal constraints.

Paper Structure

This paper contains 3 theorems, 45 equations, 1 figure.

Key Result

Lemma 1

During $\mathfrak{m}_{k\text{eq}}$-controlled spontaneous evolution of $\mathfrak{m}_{k}$ of $\bar{\Sigma}$ in Gen-GSL-Th, $\mathfrak{m}_{k}$ performs nonnegative microwork $\Delta W_{k}$ as $\mathfrak{m}_{k}\rightarrow\mathfrak{m}_{k}^{^{\prime}}$. Performing ensemble average with arbitrary $\left\

Figures (1)

  • Figure 1: Schematic forms of microenergy $E_{k}$ (dashed-dot curves) and macrowork function $E_{\text{w}}$ (solid curves) as functions of the internal variable $\boldsymbol{\xi}$, with $\xi=0$ denoting EQ in $\mathfrak{S}_{\mathbf{Z}}$. Alternatively, these curves can be considered as a function of time $t$ in $\mathfrak{S}_{\mathbf{X}}$, which increases along the directions of the blue and red arrows. We only consider the case when each curve has a single extremum. The discussion is easily extended to more complex forms. The blue color curves and solid blue arrows represent the evolution controlled by the stable (s) case. The red color curves and solid red arrows represent the evolution controlled by the unstable (u) case. In both cases, the arrows lower the energy. The extrema of all curves occur at $\boldsymbol{\xi}=0$ and represent a uniform body. For the extremum to denote equilibrium, we must also have $\overset{\cdot}{\boldsymbol{\xi}}=0$ there. The green double-arrow is discussed in the text.

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 2
  • Theorem 3