2 5 Ja n 20 06 Ludwig Boltzmann , Transport Equation and the Second Law 1

Ludwig Boltzmann had a hunch that irreversibility exhibited by a macroscopic system arises from the reversible dynamics of its microscopic constituents. He derived a nonlinear integro-differential equation-now called the Boltzmann equation-for the phase space density of the molecules of a dilute fluid. He showed that the Second law of thermodynamics emerges from Newton's equations of motion. However Boltzmann realized that stosszahlansatz, employed in the derivation, smuggles in an element of stochasticity into the transport equation. He then proposed a fully stochastic description of entropy which laid the foundation for statistical mechanics. Recent developments, embodied in different fluctuation theorems, have shown that Boltzmann's hunch was, in essence, correct. Boltzmann transport equation has played an important role in basic and applied sciences. It is a nonlinear integro-differential equation for the phase space density of the molecules of a dilute gas. It remains today, an important theoretical technique for investigating nonequilibrium systems. It was derived by Ludwig Eduard Boltzmann (1844-1906) in his further studies on thermal equilibrium between gas molecules [1], published in the year 1872. Boltzmann did this work solely for purpose of addressing the conflict between time reversal invariant Newtonian mechanics and time arrowed thermodynamics. Linear version of this equation [2] provides an exact description of neutron transport in nuclear reactor core and shields. Linear transport equation constitutes the backbone of nuclear industry. It is indeed appropriate that the Indian Society for Radiation Physics (ISRP) has chosen Boltzmann transport equation as focal theme for the sixteenth National Symposium on Radiation Physics, NSRP-16. Incidentally the year 2006 marks the hundredth anniversary of Boltzmann's death. There are going to be several talks [3] in this symposium, covering various aspects of linear transport equation. However, in this opening talk, I shall deal with nonlinear transport equation. I shall tell you of Boltzmann's lifelong struggle for comprehending the mysterious emergence of time asymmetric behaviour in a macroscopic object from the time symmetric behaviour of its microscopic constituents. This is called the Second law. In the synthesis of a macro from its micro, why and when does time reversal invariance break down? This is a question that troubled the scientists then and which troubles us now. In the early days, physicists felt that the Second law could not be derived from Newton's equations of motion. They felt the Second law must be a consequence of our inability to track large number of molecules.