Jarzynski Equality and its Special Trajectory Ensemble Average Demystified
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 deEk but neglects the ubiquitous change diEk due to external-internal force imbalance, though diEk'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.