statistical mechanics: quantitative study of systems consisting of a large number of interacting elements, such as the atoms or molecules of a solid, liquid, or gas, or the individual quanta of light (see photon photon (fō`tŏn), the particle composing light and other forms of electromagnetic radiation , sometimes called light quantum.
..... Click the link for more information. ) making up electromagnetic radiation. Although the nature of each individual element of a system and the interactions between any pair of elements may both be well understood, the large number of elements and possible interactions can present an almost overwhelming challenge to the investigator who seeks to understand the behavior of the system. Statistical mechanics provides a mathematical framework upon which such an understanding may be built. Since many systems in nature contain large number of elements, the applicability of statistical mechanics is broad. In contrast to thermodynamics Carnot cycle after the French physicist Sadi Carnot , who first discussed the implications of such cycles. During the Carnot cycle occurring in the operation of a heat engine, a definite quantity of heat is absorbed from a reservoir at high temperature; part of this heat is
..... Click the link for more information. , which approaches such systems from a macroscopic, or large-scale, point of view, statistical mechanics usually approaches systems from a microscopic, or atomic-scale, point of view. The foundations of statistical mechanics can be traced to the 19th-century work of Ludwig Boltzmann, and the theory was further developed in the early 20th cent. by J. W. Gibbs. In its modern form, statistical mechanics recognizes three broad types of systems: those that obey Maxwell-Boltzmann statistics, those that obey Bose-Einstein statistics Bose-Einstein statistics, class of statistics that applies to elementary particles called bosons, which include the photon , pion , and the W and Z particles .
..... Click the link for more information. , and those that obey Fermi-Dirac statistics Fermi-Dirac statistics, class of statistics that applies to particles called fermions. Fermions have half-integral values of the quantum mechanical property called spin and are "antisocial" in the sense that two fermions cannot exist in the same state.
..... Click the link for more information. . Maxwell-Boltzmann statistics apply to systems of classical particles, such as the atmosphere, in which considerations from the quantum theory quantum theory, modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.
..... Click the link for more information. are small enough that they may be ignored. The other two types of statistics concern quantum systems: systems in which quantum-mechanical properties cannot be ignored. Bose-Einstein statistics apply to systems of bosons (particles that have integral values of the quantum mechanical property called spin); an unlimited number of bosons can be placed in the same state. Photons, for instance, are bosons, and so the study of electromagnetic radiation, such as the radiation of a black body black body, in physics, an ideal black substance that absorbs all and reflects none of the radiant energy falling on it. Lampblack, or powdered carbon, which reflects less than 2% of the radiation falling on it, approximates an ideal black body.
..... Click the link for more information. involves the use of Bose-Einstein statistics. Fermi-Dirac statistics apply to systems of fermions (particles that have half-integral values of spin); no two fermions can exist in the same state. Electrons are fermions, and so Fermi-Dirac statistics must be employed for a full understanding of the conduction of electrons in metals. Statistical mechanics has also yielded deep insights in the understanding of magnetism magnetism, force of attraction or repulsion between various substances, especially those made of iron and certain other metals; ultimately it is due to the motion of electric charges.
..... Click the link for more information. , phase transitions, and superconductivity superconductivity, abnormally high electrical conductivity of certain substances. The phenomenon was discovered in 1911 by Kamerlingh Onnes, who found that the resistance of mercury dropped suddenly to zero at a temperature of about 4.2°K;.
..... Click the link for more information. .
statistical mechanics [stə′tis·tə·kəl mi′kan·iks]
Statistical mechanics
Statistical mechanics gives more than an explanation of already known phenomena. By using statistical methods, it often becomes possible to obtain expressions for empirically observed parameters, such as viscosity coefficients, heat conduction coefficients, and virial coefficients, in terms of the forces between molecules. Statistical considerations also play a significant role in the description of the electric and magnetic properties of materials. See Boltzmann statistics, Intermolecular forces, Kinetic theory of matter
If the problem of molecular structure is attacked by statistical methods, the contributions of internal rotation and vibration to thermodynamic properties, such as heat capacity and entropy, can be calculated for models of various proposed structures. Comparison with the known properties often permits the selection of the correct molecular structure.
Perhaps the most dramatic examples of phenomena requiring statistical treatment are the cooperative phenomena or phase transitions. In these processes, such as the condensation of a gas, the transition from a paramagnetic to a ferromagnetic state, or the change from one crystallographic form to another, a sudden and marked change of the whole system takes place. See Phase transitions
Statistical considerations of quite a different kind occur in the discussion of problems such as the diffusion of neutrons through matter. In this case, the probability of the various events which affect the neutron are known, such as the capture probability and scattering cross section. The problem here is to describe the physical situation after a large number of these individual events. The procedures used in the solution of these problems are very similar to, and in some instances taken over from, kinetic considerations. Similar problems occur in the theory of cosmic-ray showers.
It happens in both low-energy and high-energy nuclear physics that a considerable amount of energy is suddenly liberated. An incident particle may be captured by a nucleus, or a high-energy proton may collide with another proton. In either case, there is a large number of ways (a large number of degrees of freedom) in which this energy may be utilized. To survey the resulting processes, one can again invoke statistical considerations. See Scattering experiments (nuclei)
Of considerable importance in statistical physics are the random processes, also called stochastic processes or sometimes fluctuation phenomena. The brownian motion, the motion of a particle moving in an irregular manner under the influence of molecular bombardment, affords a typical example. The stochastic processes are in a sense intermediate between purely statistical processes, where the existence of fluctuations may safely be neglected, and the purely atomistic phenomena, where each particle requires its individual description. See Brownian movement
All statistical considerations involve, directly or indirectly, ideas from the theory of probability of widely different levels of sophistication. The use of probability notions is, in fact, the distinguishing feature of all statistical considerations.
Observations made on a system always require a finite time; during this time the microscopic details of the system will generally change considerably as the phase point moves. The result of a measurement of a quantity Q will therefore yield the time average, as in Eq. (1). The integral is along the trajectory
Ensembles. J. Willard Gibbs first suggested that instead of calculating a time average for a single dynamical system, a collection of systems, all similar to the original one, should instead be considered. Such an ensemble of systems is to be constructed in harmony with the available knowledge of the single system, and may be represented by an assembly of points in the phase space, each point representing a single system. If, for example, the energy of a system is precisely known, but nothing else, the appropriate representative example would be a uniform distribution of ensemble points over the energy surface, and no ensemble points elsewhere. An ensemble is characterized by a density function ρ(x1, …,zN; px1, …,pzN;t) ≡ p(x,p,t). The significance of this function is that the number of ensemble systems dNe contained in the volume element dx1 … dzN; dpx … dpzN of the phase space (this volume element will be called dΓ) at time t is as given in Eq. (2).
Relation to thermodynamics. It is certainly reasonable to assume that the appropriate ensemble for a thermodynamic equilibrium state must be described by a density function which is independent of the time, since all the macroscopic averages which are to be computed as ensemble averages are time-independent.
The so-called microcanonical ensemble is defined by Eq. (4a), where c is a constant, for the energy E between E0 and E0 + ΔE; for other energies Eq. (4b)
There is yet another ensemble which is extremely useful and which is particularly suitable for quantum-mechanical applications. Much work in statistical mechanics is based on the use of this so-called grand canonical ensemble. The grand ensemble describes a collection of systems; the number of particles in each system is no longer the same, but varies from system to system. The density function p(N,p,x) dΓN gives the probability that there will be in the ensemble a system having N particles, and that this system, in its 6N-dimensional phase space ΓN, will be in the region of phase space dΓN.
Héctor A. Chacón C.
..... Click the link for more information. ) making up electromagnetic radiation. Although the nature of each individual element of a system and the interactions between any pair of elements may both be well understood, the large number of elements and possible interactions can present an almost overwhelming challenge to the investigator who seeks to understand the behavior of the system. Statistical mechanics provides a mathematical framework upon which such an understanding may be built. Since many systems in nature contain large number of elements, the applicability of statistical mechanics is broad. In contrast to thermodynamics Carnot cycle after the French physicist Sadi Carnot , who first discussed the implications of such cycles. During the Carnot cycle occurring in the operation of a heat engine, a definite quantity of heat is absorbed from a reservoir at high temperature; part of this heat is
..... Click the link for more information. , which approaches such systems from a macroscopic, or large-scale, point of view, statistical mechanics usually approaches systems from a microscopic, or atomic-scale, point of view. The foundations of statistical mechanics can be traced to the 19th-century work of Ludwig Boltzmann, and the theory was further developed in the early 20th cent. by J. W. Gibbs. In its modern form, statistical mechanics recognizes three broad types of systems: those that obey Maxwell-Boltzmann statistics, those that obey Bose-Einstein statistics Bose-Einstein statistics, class of statistics that applies to elementary particles called bosons, which include the photon , pion , and the W and Z particles .
..... Click the link for more information. , and those that obey Fermi-Dirac statistics Fermi-Dirac statistics, class of statistics that applies to particles called fermions. Fermions have half-integral values of the quantum mechanical property called spin and are "antisocial" in the sense that two fermions cannot exist in the same state.
..... Click the link for more information. . Maxwell-Boltzmann statistics apply to systems of classical particles, such as the atmosphere, in which considerations from the quantum theory quantum theory, modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.
..... Click the link for more information. are small enough that they may be ignored. The other two types of statistics concern quantum systems: systems in which quantum-mechanical properties cannot be ignored. Bose-Einstein statistics apply to systems of bosons (particles that have integral values of the quantum mechanical property called spin); an unlimited number of bosons can be placed in the same state. Photons, for instance, are bosons, and so the study of electromagnetic radiation, such as the radiation of a black body black body, in physics, an ideal black substance that absorbs all and reflects none of the radiant energy falling on it. Lampblack, or powdered carbon, which reflects less than 2% of the radiation falling on it, approximates an ideal black body.
..... Click the link for more information. involves the use of Bose-Einstein statistics. Fermi-Dirac statistics apply to systems of fermions (particles that have half-integral values of spin); no two fermions can exist in the same state. Electrons are fermions, and so Fermi-Dirac statistics must be employed for a full understanding of the conduction of electrons in metals. Statistical mechanics has also yielded deep insights in the understanding of magnetism magnetism, force of attraction or repulsion between various substances, especially those made of iron and certain other metals; ultimately it is due to the motion of electric charges.
..... Click the link for more information. , phase transitions, and superconductivity superconductivity, abnormally high electrical conductivity of certain substances. The phenomenon was discovered in 1911 by Kamerlingh Onnes, who found that the resistance of mercury dropped suddenly to zero at a temperature of about 4.2°K;.
..... Click the link for more information. .
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statistical mechanics
Branch of physics that combines the principles and procedures of statistics with the laws of both classical mechanics and quantum mechanics. It considers the average behaviour of a large number of particles rather than the behaviour of any individual particle, drawing heavily on the laws of probability, and aims to predict and explain the measurable properties of macroscopic (bulk) systems on the basis of the properties and behaviour of their microscopic constituents.
For more information on statistical mechanics, visit Britannica.com. Britannica Concise Encyclopedia. Copyright © 1994-2008 Encyclopædia Britannica, Inc.
statistical mechanics [stə′tis·tə·kəl mi′kan·iks]
(physics)
That branch of physics which endeavors to explain and predict the macroscopic properties and behavior of a system on the basis of the known characteristics and interactions of the microscopic constituents of the system, usually when the number of such constituents is very large. Also known as statistical thermodynamics.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
Statistical mechanics
That branch of physics which endeavors to explain the macroscopic properties of a system on the basis of the properties of the microscopic constituents of the system. Usually the number of constituents is very large. All the characteristics of the constituents and their interactions are presumed known; it is the task of statistical mechanics (often called statistical physics) to deduce from this information the behavior of the system as a whole.
Scope
Elements of statistical mechanical methods are present in many widely separated areas in physics. For instance, the classical Boltzmann problem is an attempt to explain the thermodynamic behavior of gases on the basis of classical mechanics applied to the system of molecules.Statistical mechanics gives more than an explanation of already known phenomena. By using statistical methods, it often becomes possible to obtain expressions for empirically observed parameters, such as viscosity coefficients, heat conduction coefficients, and virial coefficients, in terms of the forces between molecules. Statistical considerations also play a significant role in the description of the electric and magnetic properties of materials. See Boltzmann statistics, Intermolecular forces, Kinetic theory of matter
If the problem of molecular structure is attacked by statistical methods, the contributions of internal rotation and vibration to thermodynamic properties, such as heat capacity and entropy, can be calculated for models of various proposed structures. Comparison with the known properties often permits the selection of the correct molecular structure.
Perhaps the most dramatic examples of phenomena requiring statistical treatment are the cooperative phenomena or phase transitions. In these processes, such as the condensation of a gas, the transition from a paramagnetic to a ferromagnetic state, or the change from one crystallographic form to another, a sudden and marked change of the whole system takes place. See Phase transitions
Statistical considerations of quite a different kind occur in the discussion of problems such as the diffusion of neutrons through matter. In this case, the probability of the various events which affect the neutron are known, such as the capture probability and scattering cross section. The problem here is to describe the physical situation after a large number of these individual events. The procedures used in the solution of these problems are very similar to, and in some instances taken over from, kinetic considerations. Similar problems occur in the theory of cosmic-ray showers.
It happens in both low-energy and high-energy nuclear physics that a considerable amount of energy is suddenly liberated. An incident particle may be captured by a nucleus, or a high-energy proton may collide with another proton. In either case, there is a large number of ways (a large number of degrees of freedom) in which this energy may be utilized. To survey the resulting processes, one can again invoke statistical considerations. See Scattering experiments (nuclei)
Of considerable importance in statistical physics are the random processes, also called stochastic processes or sometimes fluctuation phenomena. The brownian motion, the motion of a particle moving in an irregular manner under the influence of molecular bombardment, affords a typical example. The stochastic processes are in a sense intermediate between purely statistical processes, where the existence of fluctuations may safely be neglected, and the purely atomistic phenomena, where each particle requires its individual description. See Brownian movement
All statistical considerations involve, directly or indirectly, ideas from the theory of probability of widely different levels of sophistication. The use of probability notions is, in fact, the distinguishing feature of all statistical considerations.
Methods
For a system of N particles, each of the mass m, contained in a volume V, the positions of the particles may be labeled x1, y1, z1, …, xN, yN, zN, their cartesian velocities vx1, …, vzN, and their momenta Px1, …, PzN. This simplest statistical description concentrates on a discussion of the distribution function f(x,y,z;vx,vy,vz;t). The quantity f(x,y,z;vx,vy,vz;t) ċ (dxdydzdvxdvydvz) gives the (probable) number of particles of the system in those positional and velocity ranges where x lies between x and x + dx; vx between vx and vx + dvx, and so on. These ranges are finite.Observations made on a system always require a finite time; during this time the microscopic details of the system will generally change considerably as the phase point moves. The result of a measurement of a quantity Q will therefore yield the time average, as in Eq. (1). The integral is along the trajectory
(1) 
in phase space; Q depends on the variables x1, …, PzN, and t. To evaluate the integral, the trajectory must be known, which requires the solution of the complete mechanical problem. Ensembles. J. Willard Gibbs first suggested that instead of calculating a time average for a single dynamical system, a collection of systems, all similar to the original one, should instead be considered. Such an ensemble of systems is to be constructed in harmony with the available knowledge of the single system, and may be represented by an assembly of points in the phase space, each point representing a single system. If, for example, the energy of a system is precisely known, but nothing else, the appropriate representative example would be a uniform distribution of ensemble points over the energy surface, and no ensemble points elsewhere. An ensemble is characterized by a density function ρ(x1, …,zN; px1, …,pzN;t) ≡ p(x,p,t). The significance of this function is that the number of ensemble systems dNe contained in the volume element dx1 … dzN; dpx … dpzN of the phase space (this volume element will be called dΓ) at time t is as given in Eq. (2).
(2)
The ensemble average of any quantity Q is given (3)
by Eq. (3). The basic idea now is to replace the time average of an individual system by the ensemble average, at a fixed time, of the representative ensemble. Stated formally, the quantity defined by Eq. (1), in which no statistics is involved, is identified with defined by Eq. (3), in which probability assumptions are explicitly made. Relation to thermodynamics. It is certainly reasonable to assume that the appropriate ensemble for a thermodynamic equilibrium state must be described by a density function which is independent of the time, since all the macroscopic averages which are to be computed as ensemble averages are time-independent.
The so-called microcanonical ensemble is defined by Eq. (4a), where c is a constant, for the energy E between E0 and E0 + ΔE; for other energies Eq. (4b)
(4{\it a})
(4{\it b})
holds. By using Eq. (3), any microcanonical average may be calculated. The calculations, which involve integrations over volumes bounded by two energy surfaces, are not trivial. Still, many of the results of classical Boltzmann statistics may be obtained in this way. For applications and for the interpretation of thermodynamics, the canonical ensembles is much more preferable. This ensemble describes a system which is not isolated but which is in thermal contact with a heat reservoir. There is yet another ensemble which is extremely useful and which is particularly suitable for quantum-mechanical applications. Much work in statistical mechanics is based on the use of this so-called grand canonical ensemble. The grand ensemble describes a collection of systems; the number of particles in each system is no longer the same, but varies from system to system. The density function p(N,p,x) dΓN gives the probability that there will be in the ensemble a system having N particles, and that this system, in its 6N-dimensional phase space ΓN, will be in the region of phase space dΓN.
McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.
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