Jarzynski Equality and its Special Trajectory Ensemble Average Demystified

TL;DR

This paper clarifies misconceptions about the Jarzynski Equality, showing that the special trajectory ensemble average does not directly support the inequality as a nonequilibrium result and that JE lacks built-in irreversibility.

Contribution

It demonstrates that the common interpretation of the Jarzynski Equality's Jensen inequality is incorrect and highlights the neglect of internal energy changes in the microscopic treatment.

Findings

<R>_0 > delta(F) even in reversible processes

<R> and <R>_0 are different averages

JE does not inherently include irreversibility

Abstract

The special trajectory ensemble average (TEA), denoted by a subscript 0, in the Jarzynski Equality (JE) results in the Jensen inequality <R>_0 GT-EQ delta(F) for the work R done on the system, and not the thermodynamic work inequality <R> GT-EQ delta(F) since we find <R> NEQ <R>_0. Therefore, contrary to the common belief, the Jensen inequality does not directly support the JE as a nonequilibrium result. Jarzynski's microscopic treatment of the inclusive energy considers only the external work d_eE_k but neglects the ubiquitous change d_iE_k due to external-internal force imbalance, though d_iE_k's are present even in a reversible process as we show. Because of this neglect, no thermodynamic force necessary for dissipation is allowed. Thus the JE has no built-in irreversibility, despite a time-dependent work protocol. We support our claim by an explicit calculation, which shows that <R>_0 > delta(F) even for a reversible process for which <R> = delta(F). This also confirms that <R> and <R>_0 are different averages.