Statistical Mechanics May Have Dynamic Future

One may raise the question whether statistical mechanics, which deals mainly with large equilibrium systems, their average properties, and their small parameter fluctuations, will survive into the 21st century in its present form. If it does not survive, what kinds of changes in its conceptual machinery and mathematical description of the equilibrium systems can be expected in the current century?

Appropriate time scales for the survival of statistical mechanics may be assumed by the fact that its basic concepts have not changed substantially for about 100 years, during which time the field has demonstrated a tremendous success. Of course, the relativistic and quantum revolutions have introduced new important ideas and methods. ^1–3

The conceptual foundations of statistical mechanics and the mathematical description of equilibrium systems will likely experience substantial qualitative changes in the 21st century, changes that can be extended to nonequilibrium systems. In this regard, I offer a few guesses about some general directions in which statistical physics may change.

Statistical physics is expected to describe explicitly the stochastic dynamics of small and large random (in time and space) fluctuations of energy and other parameters. Fluctuations of various time and space scales that exist permanently in any equilibrium system reveal that an underworld of agitated, ever-changing microscopic processes exists behind even the most quiescent-appearing macroscopic states. Describing the stochastic dynamics of those fluctuations will require new mathematical tools different from those currently used. One can expect these tools to include multidimensional random functions, especially in systems of interacting particles, and other aspects of the theory of random functions. As a result, statistical physics describing time-dependent fluctuations in equilibrium systems is anticipated to acquire dynamical character.

I also expect that statistical physics will be able to describe the dynamics of both small fluctuations and large short-lived fluctuations (LSLFs) of energy and other parameters. Individual random LSLFs, which are most likely to emerge in small regions, generate peak deviations of the fluctuating parameters (energy and so forth) much greater than mean values of these parameters. Sequences of great random numbers of LSLFs appear permanently in equilibrium systems. Their inclusion in statistical physics requires description of two kinds of substantially different phenomena of finite duration. The first is LSLF formation associated with advanced processes that create within a small region a large, short-term peak deviation of the fluctuating parameters from their mean values. Thus the advanced processes precede the peak. The second kind is LSLF relaxation associated with retarded processes following the peak. In condensed matter, these processes include not only the spontaneous energy enhancement (and reduction) of one or a few strongly fluctuating particles (SFPs) during the finite time but also the accompanying correlated motion of the SFPs’ surroundings. The correlated many-body processes involving SFPs and their surroundings will require mathematical tools that take into account the finite “memories” in time and space that characterize LSLF dynamics.

Novel results of fundamental and practical importance will likely develop from the study of LSLF dynamics. Large energy fluctuations (LEF) of atoms and molecules have a critical role in a broad range of rate processes during which LEF-generated fluctuating particles overcome high energy barriers. These processes are studied and applied in physics, chemical physics, material sciences and engineering, and other related fields. The absence of a proper LEF theory has led to large discrepancies between the conventional theory of rate processes in condensed matter and experimental observations.

More than 50 years ago, Jacob Frenkel ⁴ emphasized the need to develop a many-body theory of LEFs of atomic particles in condensed matter; such a theory should include strong interactions between atoms. Frenkel also stressed difficulties in developing such theory and a theory of LEF-induced rate processes. His comments seem applicable to the present situation in standard statistical physics of condensed matter; the field presently lacks the many-body theory of LEFs of small numbers of particles. Consequently, many large discrepancies between the traditional rate theory and experimental data can still be found in the literature.

Attempts to develop such a theory have been undertaken in the stochastic kinetic many-body theory of shortlived (picosecond) large energy fluctuations (SLEFs) in nanometer regions of solids. ⁵ Encouraging results obtained in the SLEF theory and its applications suggest that these guesses on future trends have a solid basis. The future dynamic statistical physics, which will include the explicit stochastic dynamic treatment of both large and small random fluctuations, is expected to substantially reduce theory-experiment discrepancies and discover novel unusual phenomena.