Page 15 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena
P. 15

Introduction
















            The problems of statistical mechanics are those which involve systems with a
            larger number of degrees of freedom than we can conveniently follow explicitly
            in experiment, theory or simulation. The number of degrees of freedom which can
            be followed explicitly in simulations has been changing very rapidly as computers
            and algorithms improve. However, it is important to note that, even if computers
            continue to improve at their present rate, characterized by Moore’s “law,” scientists
            will not be able to use them for a very long time to predict many properties of nature
            by direct simulation of the fundamental microscopic laws of physics. This point is
            important enough to emphasize.
              Suppose that, T years from the present, a calculation requiring computation time
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            t 0 at present will require computation time t(T ) = t 0 2 −T/2  (Moore’s “law,” see
            Figure 1). Currently, state of the art numerical solutions of the Schr¨odinger equation
            for a few hundred atoms can be carried out fast enough so that the motion of these
            atoms can be followed long enough to obtain thermodynamic properties. This is
            adequate if one wishes to predict properties of simple homogeneous gases, liquids
            or solids from first principles (as we will be discussing later). However, for many
            problems of current interest, one is interested in entities in which many more atoms
            need to be studied in order to obtain predictions of properties at the macroscopic
            level of a centimeter or more. These include polymers, biomolecules and nanocrys-
            talline materials for example. In such problems, one easily finds situations in which
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            a first principles prediction requires following 10 atoms dynamically. The first
            principles methods for calculating the properties increase in computational cost as
            the number of atoms to a power between 2 and 3. Suppose they scale as the second
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            power so the computational time must be reduced by a factor 10 in order to handle
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            10 atoms. Using Moore’s law we then predict that the calculation will be possible
            T years from the present where T = 16/log 2 = 53 years. In fact, this may be
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            optimistic because Moore’s “law” may not continue to be valid for that long and
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            also because 10 atoms will not be enough in many cases. What this means is that,
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