Journal of Statistical Physics, VoL 42, Nos. 5/6, 1986 A Geometrical Measure for Entropy Changes Tova Feldmann, 1 R. D. Levine,1 and Peter Salamon 2 Received September 4, 1984; revision received September 6, 1985 The geometrical approach to statistical mechanics is used to discuss changes in entropy upon sequential displacements of the state of the system. An inter- pretation of the angle between two states in terms of entropy differences is thereby provided. A particular result of note is that any state can be resolved into a state of maximal entropy (both states having the same expectation values for the constraints) and an orthogonal component. A cosine law for the general case is also derived. KEY WORDS: Entropy changes; statistical mechanics. 1. INTRODUCTION The geometrical approach to thermodynamics(l~ has centered attention on equilibrium states. More recently, it has been applied to processes (z3~ and has been generalized to systems not in equilibrium. (4 8) Our intention here is to consider a general process through a sequence of arbitrary states and relate the fundamental new notion of the geometrical approach, namely, the angle between two states, to the change in entropy. When we adopt a statistical description where the physical state of the system is given uniquely by specifying a probability distribution {p~, i= 1,..., N} over the N possible, mutually exclusive, and collectively exhaustive states. In the geometrical approach the scalar product of two states is given by p'q=~ go.p"q j (1) i j Department of Physical Chemistry, The Hebrew University, Jerusalem 91904, Israel. 2 Department of Mathematical Sciences, San Diego State University, San Diego, California 92182. 1127 0022-4715/86/030%1127505.00/0 9 1986 Plenum Publishing Corporation