Monte Carlo methods in statistical physics: Mathematical foundations and strategies
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Your email. Send Cancel. Check system status. Toggle navigation Menu. Name of resource. Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. A guide to Monte Carlo simulations in statistical physics. Responsibility David P. Landau, Kurt Binder. Edition Fourth edition. Physical description xvii, pages : ill. Online Available online. Cambridge Core Full view. The Hamiltonian for this model can be written as:. During the simulation, we select lattice sites randomly and choose integers between [1 : q ] randomly for new Potts spin values.
Actually one of these two conditions can be used to get the absolute density of states and the other condition to check the accuracy of the result. To guarantee the accuracy of thermodynamic quantities at low temperatures, we use the condition that the number of the ground states is q to normalize the density of states. Conventional Monte Carlo simulation e. Metropolis sampling [1, 2] determines the canonical distribution P E, T by generating a random walk Markov chain at a given temperature:. From the simulational result for g E , we can calculate the canonical distribution by the above formula at essentially any temperature without performing multiple simulations.
The ''transition temperatures" T c L are determined by the temperatures where the double peaks are of the same height. Note that the peaks of the distributions are normalized to 1 in this figure. The valley between two peaks is quite deep. The latent heat for this temperature driven first-order phase transition can be estimated from the energy difference between the double peaks. Our results for the locations of the peaks are consistent with the results obtained by multicanonical method  and multibondic cluster algorithm  for those lattice sizes for which these other methods are able to generate estimates.
Wang-Landau sampling also produces results for substantially larger systems than have been studied by these other approaches. Because of the double peak structure at a first-order phase transition, conventional Monte Carlo simulations are not efficient since an extremely long time is required for the system to travel from one peak to the other in energy space. With ''Wang-Landau sampling" all possible energy levels are visited with equal probability, so it overcomes the tunneling barrier between the coexisting phases in the conventional Monte Carlo simulations.
If we are only interested in a specific temperature range, e. The inset of Fig. If we need to know the density of states more accurately for some energies, we also can perform separate simulations, one for low energy levels, one for high energy levels, the other for middle energy which includes double peaks of the canonical distribution at T c.
This scheme not only speeds up the simulation, but also increases the probability of accessing the energy levels for which both maximum and minimum values of the distributions occur by performing the random walk in a relatively small energy range. If we perform single random walk over all possible energies, it will take a long time to generate rare spin configurations. Such rare energy levels include the ground energy level or low energy levels with only few spins with different values and high energy levels where all, or most, adjacent Potts spins have different values.
However, for a larger system it is advisable to accelerate the calculation by performing random walks in different energy regions, each using a different processor. The histograms from the individual random walks are shown in the inset of Fig. In this case, we only require that the histogram of the random walk in the corresponding energy segment is sufficiently flat without regard to the relative flatness over the entire energy range.
According to finite-size scaling theory, the ''effective" transition temperature for finite systems behaves as:. We also compared our results with the existing numerical data such as estimates of transition temperatures and double peak locations obtained with the multicanonical simulational method by Berg and Neuhaus  and the Multibondic cluster algorithm by Janke and Kappler .
Monte Carlo Methods in Statistical Physics
Using the multicanonical method and a finite scaling guess for the density of states, Berg et. When we calculate the internal energy near the transition, we find that a very sharp ''jump" appears in the internal energy at T c. The magnitude of this jump equals the latent heat for the first-order phase transition and is related to the double peak distribution of the first-order phase transition.
When T is slightly different than T c , one of the double peaks increases dramatically in magnitude and the other decreases. Since we only perform simulations on finite lattices, and use a continuous function to calculate thermodynamic quantities, all our quantities for finite-size systems will appear to be continuous if we use a very small scale. The same density of states can be used to calculate the internal energy for temperatures very close to T c.
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On this scale the ''discontinuity'' at the first-order phase transition disappears and a smooth curve can be seen instead of a sharp ''jump''. When the system size goes to infinity, the discontinuity will be real. The specific heat in the vicinity of the transition temperature T c shows pronounced finite size behavior.
We find it has a finite maximum value for a given lattice size L that, according to finite-size scaling theory for first-order transitions, should vary as:. The results for the free energy per lattice site are shown in Fig. Since the transition is first-order the free energy appears to have a ''discontinuity'' in the first derivative at T c.
This is typical behavior for a first-order phase transition, and even with the fine scale used in the inset of Fig. The transition temperature T c is determined by the point where the first derivative appears to be discontinuous.
With a coarse temperature scale we can not distinguish the finite-size behavior of our model; however, we can see a very clear size dependence when we view the free energy on a very fine scale as in the inset. The entropy is another very important thermodynamic quantity that cannot be calculated directly in conventional Monte Carlo simulations.
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It can be estimated by integrating over other thermodynamic quantities, such as specific heat, but the result is not always reliable since the specific heat itself is not easy to determine accurately, particularly considering the ''divergence" at the first-order transition. With an accurate density of states estimated by our method, we already know the Gibbs free energy and internal energy for the system, so the entropy can be calculated easily:.
The entropy has a very sharp ''jump'' at T c , just as does the internal energy, and this is easily visible in a plot of the entropy near T c. The change of the entropy at T c can be obtained by the latent heat divided by the transition temperature, and the latent heat can be obtained by the jump in internal energy at T c. With the histogram reweighting method , it is possible to use simulational data at specific temperatures to obtain complete thermodynamic information near, or between, those temperatures. Unfortunately it is usually quite hard to get accurate information in the region far away from the simulated temperature due to difficulties in obtaining good statistics, especially for large systems where the canonical distributions are very narrow.
With ''Wang-Landau" sampling, the histogram is ''flat" for the random walk and we have essentially the same statistics for all energy levels. Since the output of the simulation is the density of states, which does not depend on the temperature at all, we can then calculate most thermodynamic quantities at any temperature without repeating the simulation. We also believe the algorithm is especially useful for obtaining thermodynamic information at low temperature or at the transition temperature for the systems where the conventional Monte Carlo algorithm is not so efficient. It is straightforward to apply Wang-Landau sampling to systems with a rough energy landscape, i.
Examples are spin glasses  in which the interactions between the magnetic moments produce frustration which in turn produces a complex energy landscape. One of the simplest spin-glass models is the Edwards-Anderson model  EA model ; however, because of the rough energy landscape of such disordered systems, the relaxation times of the conventional Monte Carlo simulations are very long [44, 45, 46]. Consequently, simulations can only be performed on rather small systems, and many properties of spin glasses are still unclarified .
Here, too, Wang-Landau sampling is efficient in estimating the density of states and also very aggressive in finding the ground states.
airtec.gr/images/ver-fotos/4099-buscar-ubicacion.php From a random walk in energy space, we can estimate the ground state energy and the density of states very easily. The absolute density of states by the condition that total number of states is 2 N. If we are only interested in the quantities directly related to the energy, such as free energy, entropy, internal energy and specific heat, one dimensional random walks in energy space will suffice. However for spin glass systems, one of the most important quantities is the order parameter .
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