Definition of the Subject
Monte Carlo simulation in statistical physics uses powerful computers to obtain information on the collective behavior of systems of manyinteracting particles, based on the general framework of classical or quantum statistical mechanics. Typically these systems are too complex to allow fora reliable treatment (i. e. with errors that can be controlled) by analytical theory. Monte Carlo simulation uses (pseudo)-random numbersgenerated also on the computer, and hence is suitable to derive estimates of probability distributions and averages derived from them. Such probabilitydistributions (such as the so‐called “canonical” distribution characterizing the equilibrium state of matter at a given temperatureand volume) are the basic objects of statistical thermodynamic s. While the latter field of physics provides a convenient formal framework, it doesin most cases not yield a convenient tool for explicit calculation, and such an approach is...
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsAbbreviations
- Classical statistical mechanics:
-
Statistical mechanics relates the macroscopic properties of matter to basic equations governing the motion of the (many!) constituents from which matter is built from. For classical statistical mechanics these equations are Newton’s laws of classical mechanics.
- Critical slowing down:
-
Divergence of the relaxation time of the model describing the dynamics of a many‑particle system when one approaches a second-order phase transition (= “critical point” in the phase diagram).
- Detailed balance principle:
-
Relation linking the transition probability for a move and the transition probability for the inverse move to the ratio of the probability for the occurrence of these two states connected by these moves in thermal equilibrium.
- Equilibrium:
-
Statistical mechanics considers “thermal equilibrium”, i. e. a many-body system in contact with a (big) heat reservoir does not take up heat from this reservoir, its macroscopic properties do not change with time, and a few global properties (like temperature, pressure, particle number) suffice to characterize the state of the system.
- Ergodicity:
-
Property that ensures that ensemble averages of statistical mechanics (taken with the proper probability distribution) agree with time averages taken along the trajectory along which the system moves through its state space.
- Finite-size scaling:
-
Theory that describes the rounding and shifting of singularities that thermodynamic properties exhibit when the state of a system changes from one phase to another in the “thermodynamic limit” (i. e., particle number \( { N \rightarrow \infty } \)).
- Importance sampling:
-
Monte Carlo method that chooses the states that are generated according to the probability distribution that one desires to realize. For example, for statistical mechanics applications, states are chosen with weights proportional to the “Boltzmann factor” \( { \{ \exp [-\text{energy of the state/temperature}] \} } \).
- Master equation:
-
Rate equation describing the “time”‐evolution of the probability that a state occurs as a function of a “time” coordinate labeling the sequence of states.
- Molecular dynamics method:
-
Simulation method for interacting many-body systems based on the numerical solution of Newton’s equations of motion of classical mechanics.
- Monte Carlo step:
-
Unit of (pseudo) time in (dynamically interpreted) importance sampling where, on the average, each degree of freedom in the system gets one chance to be changed (or “updated”).
- Quantum statistical mechanics:
-
Statistical mechanics relates the macroscopic properties of matter to basic equations governing the motion of the (many!) constituents matter is built from. For quantum statistical mechanics, this basic equation is the Schrödinger equation for the many body wavefunction. If the eigenvalue spectrum of this equation could be obtained, the canonical formalism of statistical mechanics could be straightforwardly applied; since normally this is not possible, one has to use a reformulation of the Schroedinger equation in terms of path integrals.
- Random number generator (RNG):
-
Computer subroutine to produce pseudorandom numbers that are approximately uniformly distributed in the interval from zero to unity. Approximately the subsequently generated random numbers are uncorrelated. RNG’s typically are deterministic algorithms and strictly periodic, but the period is large enough that for many applications this periodicity does not matter.
- Simple sampling:
-
Monte Carlo method that chooses states uniformly and at random from the available space.
- Thermodynamic variables:
-
Macroscopic pieces of matter (solids, liquids, gases) in thermal equilibrium can be characterized by a few state variables, “extensive” thermodynamic variables (proportional to the particle number, such as energy, volume) and “intensive” ones (independent of the particle number, such as temperature, pressure).
- Transition probability:
-
Probability that controls the move from one state to the next one in a Markovian Monte Carlo process.
Bibliography
Berg BA (2004) Markov Chain Monte Carlo Simulations and Their Statistical Analysis. World Scientific, Singapore
Binder K (1981) Critical properties from Monte Carlo coarse graining and renormalization. Phys Rev Lett 47:693–696
Binder K (1982) The Monte Carlo calculation of the surface tensions for two- and three‐dimensional lattice models. Phys Rev A 25:1699–1709
Binder K, Young D (1986) Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev Mod Phys 58:801–976
Binder K (ed) (1995) Monte Carlo and Molecular Dynamics Simulations in Polymer Science. Oxford University Press, New York
Binder K, Heermann DW (2002) Monte Carlo Simulation in Statistical Physics: An Introduction, 4th edn. Springer, Berlin
Ceperley DM (1996) Path integral Monte Carlo methods for fermions. In: Binder K, Ciccotti G (eds) Monte Carlo and Molecular Dynamics of Condensed Matter Systems. Societa Italiana di Fisica, Bologna, pp 445–482
Fisher ME (1971) The theory of critical point singularities in critical phenomena. In: Green MS (ed) Proceedings of the 1970 Enrico Fermi International School of Physics, vol 51. Academic Press, New York, pp 1–99
Frenkel D, Smit B (2002) Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn. Academic Press, San Diego
Hohenberg PC, Halperin BI (1977) Theory of dynamic critical phenomena. Rev Mod Phys 49:435–479
James F (1990) A review of pseudorandom number generators. Comp Phys Commun 60:329–344
Knuth D (1969) The Art of Computer Programming, vol 2. Addison‐Wesley, Reading
Landau DP, Binder K (2005) A Guide to Monte Carlo Simulations in Statistical Physics, 2nd edn. Cambridge Univ Press, Cambridge
Mascagni M, Srinivasan A (2000) Algorithm 806: SPRNG: A Scalable Library for Pseudorandom Number Generation. ACM Trans Math Softw 26:436–461
Montvay I, Münster G (1994) Quantum Fields on the Lattice. Cambridge Univ Press, Cambridge
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AM, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092
Privman V (ed) (1990) Finite Size Scaling and Numerical Simulation of Statistical Systems. World Scientific, Singapore
Sadiq A, Binder K (1984) Dynamics of the formation of two dimensional ordered structures. J Stat Phys 35:517–585
Stauffer D, Aharony A (1994) Introduction to Percolation Theory. Taylor and Francis, London
Suzuki M (ed) (1982) Quantum Monte Carlo Methods in Condensed Matter Physics. World Scientific, Singapore
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag
About this entry
Cite this entry
Binder, K. (2009). Monte Carlo Simulations in Statistical Physics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_337
Download citation
DOI: https://doi.org/10.1007/978-0-387-30440-3_337
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-75888-6
Online ISBN: 978-0-387-30440-3
eBook Packages: Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics