The Thermodynamical Arrow of Time

Abstract

The thermodynamical arrow of time is characterized by an increase of entropy according to the Second Law, which was postulated by Clausius in 1865 on the basis of Carnot’s process. It can be written in the form (see also (3.1’) on page 57)

$$\begin{array}{*{20}{c}} {\frac{{dS}}{{dt}} = {{{\left\{ {\frac{{dS}}{{dt}}} \right\}}}_{{ext}}} + {{{\left\{ {\frac{{dS}}{{dt}}} \right\}}}_{{\operatorname{int} }}},} \\ {with\;d{{S}_{{ext}}} = \frac{{dQ}}{T}\;and\;{{{\left\{ {\frac{{dS}}{{dt}}} \right\}}}_{{\operatorname{int} }}} \geqslant 0,} \\ \end{array}$$

(3.1)

where S is the phenomenological entropy of a bounded system, while dQ is the total inward flow of heat through its boundary. (The heat flux is usually not written as a derivative dQ/dt,since its integral would not represent a ‘function of state’ Q(t), while it does, of course, define the time-integrated flux in the actual process.) The first term of (3.1) vanishes by definition for ‘thermodynamically closed’ systems. Since the whole universe is defined as an absolutely closed system (even when infinite), its total entropy, or the mean entropy with respect to coexpanding volume elements, should according to this law evolve towards an assumed maximum — the socalled Wärmetod (heat death). The phenomenological concepts used above apply only in situations of partial equilibrium, where a local temperature is defined.