Boltzmann’s H-Theorem, Maxwell Velocity Distribution and Diffusion-Entropy Scaling

 Prof. Biman Bagchi

Solid State and Structural Chemistry Unit

Indian Institute of Science, Bangalore, India

 Thursday, June 9 at 4 PM over Microsoft Teams


Kinetic Theory of Gases (KTG) forms the backbone of our understanding of dynamical processes in nature. It was pioneered, among others, by Maxwell and Boltzmann, whose theories framed the way we think about time-dependent processes. Boltzmann became obsessed with Maxwell’s velocity distribution work and spent his life developing it further. In the process, he introduced the famous H-function, his H-Theorem, derived the relation between entropy and the number of states (the Boltzmann’s formula, which also introduces Boltzmann’s constant in science). The relationship between entropy and relaxation has been the subject of inquiry ever since the time Boltzmann made his famous remark. In this Lecture, we shall present an alternate derivation of Maxwell velocity distribution that uses the Central Limit Theorem and Boltzmann’s entropy formula, discuss the time dependence of the H-function and its possible relation with entropy. On a more modern topic, we discuss a derivation of the diffusion-entropy scaling in a two-dimensional periodic lattice, using the multidimensional rate theory. We shall discuss a two-dimensional periodic system that displays interesting coherences in trajectories, leading to a non-monotonic friction dependence of diffusion. We discuss a simple random walk model that surprisingly could capture the non-monotonicity of diffusion (partly) and the behaviour at large friction (quantitatively). We shall also discuss, if time permits, a few biophysical problems that we are studying in our group.


  1. L. Boltzmann, Wien. Ber. 66, 275 (1872).
  2. Maxwell, J. Clerk-Maxwell’s Kinetic Theory of Gases. Nature 8, 85 (1873).
  3. Chapman, S. and Cowling, T.G., 1990. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge university press


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