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In statistical mechanics, a microstate is a specific microscopic configuration of a
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
that the system may occupy with a certain probability in the course of its
thermal fluctuations In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.In statistical mechanics they are often simply referred to as fluctuations. All thermal fluctuations b ...
. In contrast, the macrostate of a system refers to its macroscopic properties, such as its
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
,
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
and
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
. Treatments on statistical mechanics define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate. A macrostate is characterized by a probability distribution of possible states across a certain
statistical ensemble In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
of all microstates. This distribution describes the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of finding the system in a certain microstate. In the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
, the microstates visited by a macroscopic system during its fluctuations all have the same macroscopic properties.


Microscopic definitions of thermodynamic concepts

Statistical mechanics links the empirical thermodynamic properties of a system to the statistical distribution of an ensemble of microstates. All macroscopic thermodynamic properties of a system may be calculated from the partition function that sums \text(-E_i/kT) of all its microstates. At any moment a system is distributed across an ensemble of \Omega microstates, each labeled by i, and having a probability of occupation p_i, and an energy E_i. If the microstates are quantum-mechanical in nature, then these microstates form a discrete set as defined by quantum statistical mechanics, and E_i is an
energy level A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
of the system.


Internal energy

The internal energy of the macrostate is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
over all microstates of the system's energy :U \,:=\, \langle E\rangle \,=\, \sum\limits_^\Omega p_i \, E_i This is a microscopic statement of the notion of energy associated with the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...
.


Entropy

For the more general case of the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat ...
, the absolute
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
depends exclusively on the probabilities of the microstates and is defined as :S \,:=\, -k_\mathrm \sum\limits_^\Omega p_i \, \ln (p_i) where k_B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. For the
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
, consisting of only those microstates with energy equal to the energy of the macrostate, this simplifies to : S = k_B\,\ln \Omega with the number of microstates \Omega = 1/p_i. This form for entropy appears on
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
's gravestone in Vienna. The
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
describes how the entropy of an isolated system changes in time. The
third law of thermodynamics The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
is consistent with this definition, since zero entropy means that the macrostate of the system reduces to a single microstate.


Heat and work

Heat and work can be distinguished if we take the underlying quantum nature of the system into account. For a closed system (no transfer of matter),
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
in statistical mechanics is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the quantum energy levels of the system, without change in the values of the energy levels themselves.
Work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...
is the energy transfer associated with an ordered, macroscopic action on the system. If this action acts very slowly, then the
adiabatic theorem The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows: :''A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it sl ...
of quantum mechanics implies that this will not cause jumps between energy levels of the system. In this case, the internal energy of the system only changes due to a change of the system's energy levels. The microscopic, quantum definitions of heat and work are the following: :\delta W = \sum_^N p_i\,dE_i :\delta Q = \sum_^N E_i\,dp_i so that :~dU = \delta W + \delta Q. The two above definitions of heat and work are among the few expressions of statistical mechanics where the thermodynamic quantities defined in the quantum case find no analogous definition in the classical limit. The reason is that classical microstates are not defined in relation to a precise associated quantum microstate, which means that when work changes the total energy available for distribution among the classical microstates of the system, the energy levels (so to speak) of the microstates do not follow this change.


The microstate in phase space


Classical phase space

The description of a classical system of ''F'' degrees of freedom may be stated in terms of a 2''F'' dimensional phase space, whose coordinate axes consist of the ''F''
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
''qi'' of the system, and its ''F'' generalized momenta ''pi''. The microstate of such a system will be specified by a single point in the phase space. But for a system with a huge number of degrees of freedom its exact microstate usually is not important. So the phase space can be divided into cells of the size ''h''0 = Δ''qi''Δ''pi'', each treated as a microstate. Now the microstates are discrete and countable and the internal energy ''U'' has no longer an exact value but is between ''U'' and ''U''+''δU'', with \delta U\ll U''.'' The number of microstates Ω that a closed system can occupy is proportional to its phase space volume: \Omega(U)=\frac\int\ \mathbf_(H(x)-U) \prod_^\mathcaldq_i dp_i where \mathbf_(H(x)-U) is an Indicator function. It is 1 if the Hamilton function ''H''(''x'') at the point ''x'' = (''q'',''p'') in phase space is between ''U'' and ''U''+'' δU'' and 0 if not. The constant / makes Ω(''U'') dimensionless. For an ideal gas is \Omega (U)\propto\mathcalU^\delta U. In this description, the particles are distinguishable. If the position and momentum of two particles are exchanged, the new state will be represented by a different point in phase space. In this case a single point will represent a microstate. If a subset of ''M'' particles are indistinguishable from each other, then the ''M!'' possible permutations or possible exchanges of these particles will be counted as part of a single microstate. The set of possible microstates are also reflected in the constraints upon the thermodynamic system. For example, in the case of a simple gas of ''N'' particles with total energy ''U'' contained in a cube of volume ''V'', in which a sample of the gas cannot be distinguished from any other sample by experimental means, a microstate will consist of the above-mentioned ''N!'' points in phase space, and the set of microstates will be constrained to have all position coordinates to lie inside the box, and the momenta to lie on a hyperspherical surface in momentum coordinates of radius ''U''. If on the other hand, the system consists of a mixture of two different gases, samples of which can be distinguished from each other, say ''A'' and ''B'', then the number of microstates is increased, since two points in which an ''A'' and ''B'' particle are exchanged in phase space are no longer part of the same microstate. Two particles that are identical may nevertheless be distinguishable based on, for example, their location. (See configurational entropy.) If the box contains identical particles, and is at equilibrium, and a partition is inserted, dividing the volume in half, particles in one box are now distinguishable from those in the second box. In phase space, the ''N''/2 particles in each box are now restricted to a volume ''V''/2, and their energy restricted to ''U''/2, and the number of points describing a single microstate will change: the phase space description is not the same. This has implications in both the Gibbs paradox and
correct Boltzmann counting Correct or Correctness may refer to: * What is true * Accurate; Error-free * Correctness (computer science), in theoretical computer science * Political correctness, a sociolinguistic concept * Correct, Indiana, an unincorporated community ...
. With regard to Boltzmann counting, it is the multiplicity of points in phase space which effectively reduces the number of microstates and renders the entropy extensive. With regard to Gibb's paradox, the important result is that the increase in the number of microstates (and thus the increase in entropy) resulting from the insertion of the partition is exactly matched by the decrease in the number of microstates (and thus the decrease in entropy) resulting from the reduction in volume available to each particle, yielding a net entropy change of zero.


See also

* Quantum statistical mechanics *
Degrees of freedom (physics and chemistry) In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedo ...
* Ergodic hypothesis * Phase space


References

{{Reflist


External links


Some illustrations of microstates vs. macrostates
Statistical mechanics