Statistical mechanics and dynamics of polymers and membranes

Abstract

When developing theories of nonequilibrium systems and constructing stochastic models, it is important to ensure that these are consistent with equilibrium statistical mechanics. We have encountered this issue in various contexts, including the Einstein relation and fluctuation-dissipation theorems (Chap. 2 and Chap. 10), Brownian ratchets (Chap. 4), and the law of mass action, and detailed balance (Chap. 3 and Chap. 5). The essential idea is that many stochastic dynamical phenomena in cell biology involve the relaxation to an equilibrium thermodynamic state, with the relaxation occurring on relevant biological time scales. It follows that in order to understand the dynamics of cellular structures such as the cytoskeleton, plasma membrane, and nucleus, it is necessary to consider the equilibrium and near-equilibrium behavior of the physical components underlying such structures. Therefore, in this chapter, we present a basic introduction to the statistical mechanics and dynamics of polymers, membranes, and polymer networks. In order to keep the material reasonably self-contained, we use the Boltzmann-Gibbs distribution and associated partition function as the starting point of our analysis. This also provides a relatively straightforward way of defining important quantities such as free energy, entropy, and the chemical potential (Chap. 1). For a more detailed and general introduction to statistical mechanics; see Refs. [178, 489]. Extensive material on polymer physics can be found in Refs. [250, 251, 775], whereas the physics of membranes is covered in Refs. [540, 782]. Finally, Ref. [87] provides a comprehensive introduction to cell mechanics.