Loschmidt's reversibility paradox led to a very fruitful discussion between Boltzmann and Loschmidt about the second law and motivated Boltzmann to work out his sta tistical interpretation of the second law in detail. To handle the reversibility paradox Boltzmann investigated the entire phase space of a dynamical system consisting of N particles or molecules. He found that the volume in the 6N dimensional phase space of the system which represents all possible values of the three coordinates and the three momentum components of each particle can be subdivided into regions corresponding to macroscopic states of the system which we shall call macrostates.

In his paper of 1877 entitled "On the relation between the second law of the me chanical theory of heat and the probability calculus with respect to the theorems on thermal equilibrium", Boltzmann now presented a probabilistic expression for the en tropy. He could show that the entropy S is proportional to the 6N-dimensional phase space volume occupied by the corresponding macrostate of an N-particle system:

It is now usually written in the notation of Max Planck

where k is the Boltzmann constant and W is the number of microstates by which the macrostate of the system can be realized. This relation has been called Boltzmann's Principle by Albert Einstein (1879-1955) in 1905 since it can be used as the foundation of statistical mechanics. It is not limited to gases but can also be applied to liquids and solid states. It can be obtained by introducing cells of finite volume in phase space as Boltzmann had already done in order to obtain a denumerable set of microstates. It implies that the entropy is proportional to the logarithm of the so- called thermodynamic probability W of the macrostate which is just the corresponding number of microstates. A macrostate is determined by a rather small number of macro scopic variables of the system such as volume, pressure and temperature. The latter two correspond to averages over microscopic variables of the system. A microstate, on the other hand, is specified by the coordinates and momenta of all molecules of the system. Due to the large number of molecules there is a very large number of difierent choices for the individual coordinates and momenta which lead to the same macrostate. It turns out, that for a large system by far the largest number of microstates corresponds to equilibrium - and quasi - equilibrium - states as we have already illustrated in the exam ple of the liquid containing a dye. The latter are states which difier very little from the equilibrium state with maximum entropy and cannot be distinguished macroscopi cally from the equilibrium state. Thus this macrostate is the state of maximal entropy and the transition from nonequilibrium to equilibrium corresponds to a transition from exceptionally unprobable nonequilibrium-states to the extremely probable equilibrium- state. In Boltzmann's statistical interpretation the second law is thus not of absolute but only of probabilistic nature. The appearance of so-called statistical fluctuations in small subsystems was predicted by Boltzmann and he recognized Brownian motion as such a phenomenon. The theory of Brownian motion has been worked out indepen dently by Albert Einstein in 1905 and by Marian von Smoluchowski. The experimental verification of these theoretical results by Jean Baptiste Perrin was important evidence for the existence of molecules.

The term Statistical Mechanics has actually been coined by the great American physicist J. Willard Gibbs (1839-1903) at a meeting of the American Association for the Advancement of Science in Philadelphia in 1884. This was one of the rare occa sions when Gibbs went to a meeting away from New Haven. He had been professor of mathematical physics at Yale University since 1871 and had served nine years without salary. Only in 1880, when he was on the verge of accepting a professoship at John Hopkins University, did his institution offer him a salary. He had realized that the pa pers of Maxwell and Boltzmann initiated a new discipline which could be applied to bodies of arbitrary complexity moving according to the laws of mechanics which were investigated statistically. In the years following 1884 he formulated a general framework for Statistical Mechanics and in 1902 published his treatise.

Gibbs started his consideration with the principle of conservation of the phase space volume occupied by a statistical ensemble of mechanical systems. He considered three types of ensembles. The so{called microcanonical ensemble of Gibbs corresponds to an ensemble of iso- lated systems which all have the same energy. Boltzmann called this ensemble "Ergo- den". In this case each member of the ensemble coresponds to a different microstate and all microstates have the same probability. The canonical ensemble of Gibbs corresponds to systems in contact with a heat bath. In this case the energy of the individual systems is allowed to uctuate around the mean value E. If E_v is the energy of an individual system v of the ensemble, its probability P_vis proportional to an exponential function linear in the energy which is nowadays often called the Boltzmann factor.

For the grandcanonical ensemble of Gibbs not only the energy but also the number of particles N_v of the individual systems is allowed to uctuate around the mean value N. If we introduce the density in 6N-dimensional phase space for an ensemble of phys ical N-particle systems

the Gibbs entropy can be written in the form

Introducing finite cells in phase space the number of microstates becomes denumerable and will be labelled by _v = 1, 2, ... W where W is the total number of microstates. The expression for the entropy then becomes

where P_v is the probability of the corresponding microstate. It has already the same form as the corresponding expression for a quantum system with discrete energy levels. We may thus use the procedure introduced by John von Neumann (1903-1957) in 1927 to determine the equilibrium distribution P_v. It can be found by demanding that the entropy becomes a maximum under certain subsidiary conditions which implies that the variation of S with respect to the P_v vanishes. For the microcanonical ensemble only the sum of all probabilities must be one and if the total number of states is W one obtains the same probability P_v = 1 / W for all microstates.