Monte Carlo Simulations in Statistical Physics

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1 Part II Monte Carlo Simulations in Statistical Physics By D.Stauffer

2 Introduction In Statistical Physics one mostly deals with thermal motion of a system of particles at nonzero temperatures. For example, in a classical ideal gas of point-like molecules each particle has an average kinetic energy equal to dkbt /2 in d dimensions. Here T is the absolute temperature and kb = 1.6 X Joule per Kelvin is Boltzmann's constant. Statistical Physics is used try to explain such laws and to predict the properties of materials consisting of many such particles; therefore, in this example the specific heat is 3N kb /2 in three dimensions if the gas consists of N particles. In most applications, the number N of particles is very large, and they influence each other by their intermolecular forces. For example, a glass of beer contains about 1025 water molecules, and if these molecules did not interact with each other, the beer would vanish by evaporation, not by drinking. These interactions are also unhealthy for theoretical physics since with interactions usually one cannot solve exactly the problem of how the molecules move and what their average energy is, because even on a computer it is not possible to store the positions and velocities of 1025 point-like molecules. (The Cray-2 supercomputer has only two Gigabytes of main memory.) Instead, one is forced to work with a much smaller number of molecules, below 106, and solve numerically the equations of motion arising from Newton's law: force equals mass times acceleration. This method is called molecular dynamics and has already been used in the first part of this book by Zabolitzky. We will not deal with this technique here; readers who want to know more are referred to the book of D.W.Heermann [1J. Instead Statistical Physics mostly uses a probabilistic (or stochastic) method: The system of interacting particles has a fluctuating energy E, because we assume that it is in contact with a heat bath. For example, some people like to study glasses of whisky cooled by ice cubes in contact with the liquid. Energy is transferred more or less randomly from the heat bath (the ice cubes) to the system of interest (the whisky), and back. Statistical Physics assumes, in agreement with more detailed theories and with experiment, that the observed equilibrium system reaches a given total energy E with a probability P(E) '" exp(-e/kbt), where exp( x) = ex is the exponential function. Thus we shuffle the particles around randomly according to this probability, not to Newton's law. For large enough systems, the results are usually the same. Experts call this second method an example of the canonical ensemble and the first method, simulated by molecular dynamics, an example of the microcanonical ensemble. They also like the word Hamiltonian instead of energy. (Most quantum effects are neglected in these lectures.)

3 70 II Monte Carlo Simulations in Statistical Physics '?S 32. "Monte Carlo Simulation" In this part I want to avoid such names and instead simulate this probabilistic method on the computer. We tell the computer to do a certain operation with probability p by calculating a random number z which lies somewhere between zero and unity. All real numbers in that interval have the same chance to agree with that random number z. Then the given operation is executed if and only if the random number z is smaller than the probability p. For example, if p = 0.1 and if we produce a million different random numbers z, then about 90 % of these random numbers will be between 0.1 and 1, and only 10 % will be between 0 and 0.1. Thus the condition z < p is fulfilled with probability p (the next chapter deals with the production of random numbers on a computer). If the reader has the time and the money and does not demand too much accuracy, he may also produce them on the roulette tables of the Mediterranean resort town of Monte Carlo. This more traditional method has given the probabilistic Monte Carlo simulation of exp(-e/kbt) its name.

4 Introduction 71 Still, if we had to store and to change three coordinates and three velocity components for each particle of a three-dimensional experiment, we would need a lot of computer time and memory just as for molecular dynamics methods. Simulations are much simpler if we assume that the particles can only sit on the sites of a big lattice; then each lattice site is in one of only two states: Occupied or empty. This simplification is ideal for computers; only one bit is needed for each site. Moreover, we usually assume that only nearest neighbors influence each other; thus on a simple cubic lattice each site feels the forces from only six neighboring sites. This model for fluids is called the lattice gas or (spin 1/2) Ising model. The above formula exp( - E / kbt) is only valid in thermal equilibrium, i.e. when the temperature of the sample is the same as that of the surrounding heat bath and when properties like the average energy no longer change with time. In recent years many growth processes and disordered systems have also been studied which are not in thermal equilibrium. The probability to occupy a site is then defined differently, and various different growth processes are possible. Related models are the so-called cellular automata. Of course, all these Monte Carlo methods for lattice models are much easier for the computer than the molecular dynamics for a fluid. Therefore the simulation of more than 1010 sites was achieved in 1986, with up to 4200 sites treated per microsecond. (The million sites barrier was broken in 1981.) But we will still have to wait before computers beat nature with its 1025 molecules in a glass of beer. Common to all these models on lattices is a simple computing trick to save computing time: Use arrays with one index, not with two or three in 2 and 3 dimensions. For example, if the variables IS store the occupation status of the sites (IS=O or = -1 for empty, IS=l for occupied) in a simple cubic lattice of L*L*L sites, then it is not practical to store the sites as IS(I,J,K) with indices from 1 to L. Instead I prefer to store them as a one-dimensional array IS(K) with K from 1 to L3. The left neighbor of site K has the index K-1, the right neighbor is K+1, the neighbor in front (in the back) is K-L (K+L), and the neighbors above and below have the index K-L*L and K+L*L. (To ensure that the sites in the top-most plane also have an upper neighbor, and the sites in the lowest plane have a lower neighbor, I often store the content ofthe lowest plane again in a buffer plane on top of the top-most physical plane. Similarly, the content of the highest physical plane is stored in an additional buffer plane below the lowest physical plane. Then the upper buffer has indices K from 1 to L2, the physical sites are numbered from L2+ 1 to L3+L2, and the lower buffer plane has sites from K=L3+L2+1 to K=L3+2L2 which agree with sites IS(L2+1) to IS(2L2).) For two dimensions, a small 3*3 lattice is then stored as with italic numbers for the buffer lines

5 72 II Monte Carlo Simulations in Statistical Physics This method automatically leads to the so-called periodic (more precisely: helical) boundary conditions where there are no surface sites and where thus a limited number of lattice sites in general simulates better the infinitely large material than a system with free surfaces. Computer time is saved because the computer no longer has to evaluate I+J*L+K*L 2 for memory element IS(I,J,K) and because one does not need six IF-statements for the boundary sites. Moreover, on vectorcomputers a loop over all IS(K) (without buffers) now has L3 elements whereas with IS(I,J,K) the loop over K has only L elements. Vector computers often do not work efficiently if the innermost loop has only a small number of elements, and L3 is larger and thus more efficient to deal with than L. This. part deals first with the lattice gas in thermal equilibrium, and then with other models like cellular automata and growth processes. We end with the Kauffman model for genetics, which was invented by a physician, not a physicist. Apart from the next chapter on random number generation it should not be necessary to read any of the previous chapters to understand the following ones. Thus the biologically inclined reader may concentrate on Eden and Kauffman models, the solid state physicist on the Ising model. For all models, however, our emphasis will be on fast Fortran simulation oflarge systems, not on elegant programming or beautiful graphics display. We let the computer work for us, and not oblige us to follow its special wishes.

Progress toward a Monte Carlo Simulation of the Ice VI-VII Phase Transition

Progress toward a Monte Carlo Simulation of the Ice VI-VII Phase Transition Christina Gower 2010 NSF/REU PROJECT Physics Department University of Notre Dame Advisor: Dr. Kathie E. Newman August 6, 2010