Thermal Fluctuations of Interfaces

Consider the fluctuations of a surface defined in the Monge representation as z = h(x, y). The area of the flat surface is denoted by A. For slowly varying fluctuations of this surface about a flat shape (h = ho, where is a constant) the additional surface free energy of the undulated interface over that of the flat one (AFg = — yA) is approximately 

Treating AFg as a Hamiltonian of the fluctuating variable the probability to find h q) with any particular value is proportional to exp[— which in our approximation is a Gaussian where all the different q modes are independent. Using the relationship between the probability distribution and the mean-square value of a fluctuating quantity (see Chapter 1), we have 

This is the mean-square value of each Fourier mode in thermal equilibrium. 

The mean-squared, real-space fluctuations of the surface about a flat profile, are given by 

By symmetry, h (p)) is independent of the point p in the c — y plane at which the fluctuation is calculated. It is important to note that the mean-square fluctuation diverges as the logarithm of the system size. For a onedimensional interface a line instead of a surface) this divergence is even more severe and increases linearly with the size of the system. (For the line, everything is similar, except that the dimensionality of the integral in Eq. (3.17) is one-dimensional.) Thus, due to reduced dimensionality of these interfaces, the thermal fluctuations can have drastic effects on how flat the interfaces really are. 

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