ABSTRACT
Simple random walks (RWs) and self-avoiding random walks (SAWs) on lattices have been used as models for the study of statistical properties of long flexible polymer chains, since shortly after the important sampling Monte Carlo method was devised [1]. Figure 1 explains the meaning of RWs, of SAWs, and of a further variant, the nonreversal random
walk (NRRW), by giving examples of such walks on the square lattice [2]. The idea is that each site of the lattice which is occupied by the walk is interpreted as an (effective) monomer, and the lattice constant (of length a) connecting two subsequent steps of the
random walk may be taken as an effective segment connecting two effective monomers. Thus, the lattice parameter a would correspond physically to the Kuhnian step length bK. For a RW, each further step is completely independent of the previous step and, hence,
the mean square end-to-end distance after N steps simply is
(1)
and the number of configurations (i.e., the partition function of the chain ZN) is (z is the coordination number of the lattice) ZN=zN
(2)
Although the RW clearly is an extremely unrealistic model of a polymer chain, it is a useful starting point for analytic calculations since any desired property can be calculated exactly. For example, for N>>1 the distribution
of the end-to-end vector is Gaussian, as it is for the freely-jointed chain (see Chapter 1), i.e., (in d=3 dimensions),
(3)
which implies that the relative mean square fluctuation of R2 is a constant,
(4)
Equation (4) shows that one encounters a serious problem in computer simulations of polymers, which is not present for many quantities in simple systems (fluids of small
molecules, Ising spin models, etc.; see Binder and Heermann [3]), namely the “lack of selfaveraging” [4]. By “selfaveraging” we mean that the relative mean square fluctuation of a property decreases to zero when the number of degrees of freedom (i.e., the number
of steps N here) increases towards infinity. This property holds, e.g., for the internal energy E in a simulation of a fluid consisting of N molecules or an Ising magnet with N
spins; the average energy is calculated as the average of the Hamiltonian, as N→∞, at least for
thermodynamic states away from phase transitions. This selfaveraging does not hold for the mean square end-to-end distance of a single polymer chain, as Eq. (4) shows for the
RW (actually this is true for more complicated polymer models as well, only the constant ∆R may differ from , e.g., ∆R≈0.70 for the SAW in d=3 [3]).
