Microstate

In statistical physics, a microstate is the complete microscopic description of a thermodynamic system . A microstate corresponds to a point in the phase space of the whole system (not that of a particle). For a classic ideal gas , the location and momentum of each particle are determined.

In contrast to the microstate, the macrostate describes the system through its averaged parameters such as temperature , pressure or magnetization . A thermodynamic system with a given macrostate now occupies different microstates of the energy with a certain probability . Many parameters of the system can be calculated from these micro-states together with their probabilities. ${\ displaystyle E_ {i}}$ ${\ displaystyle p_ {i}}$

Often some micro-states of a closed system are not distinguishable from the outside (e.g. because they have the same total energy and the same total impulse or the same total magnetization ). According to the postulate of the same a priori probabilities , each of these micro-states occurs with the same probability in thermal equilibrium . It cannot be proven, but it is the only plausible assumption, since each marking one of these states with a changed probability would mean a certain arbitrariness.

Microscopic definition of thermodynamic quantities

Statistical physics defines the thermodynamic properties of a system using an ensemble of micro-states. Each microstate can be assigned an energy and an occupation probability, which result from the properties of the microstate. With these definitions, key figures of the system can then be calculated as the mean of the micro-states ( ensemble mean ) (see also ergodic hypothesis ). Examples: ${\ displaystyle N}$ ${\ displaystyle i}$ ${\ displaystyle E_ {i}}$ ${\ displaystyle p_ {i}}$

The internal energy is the energy of the associated macrostate as an expected value over the energies of the microstates: ${\ displaystyle U}$

{\ displaystyle U \,: = \, \ langle E \ rangle \, = \, \ sum \ limits _ {i = 1} ^ {N} p_ {i} \, E_ {i}}

The entropy of the entire system depends only on the probabilities of the microstates and is defined as the expected value of the entropies of the microstates : ${\ displaystyle S}$ ${\ displaystyle S_ {i} = - k _ {\ mathrm {B}} \ ln (p_ {i})}$

{\ displaystyle S \,: = \ sum _ {i} p_ {i} S_ {i}: = \, - k _ {\ mathrm {B}} \ cdot \ sum \ limits _ {i = 1} ^ {N } p_ {i} \, \ ln (p_ {i}) = \, - k _ {\ mathrm {B}} \ cdot \ langle \ ln (p_ {i}) \ rangle}

Where is the Boltzmann constant . This definition corresponds (without ) to Shannon's information entropy

{\ displaystyle k _ {\ mathrm {B}}}

{\ displaystyle k _ {\ mathrm {B}}}

Further thermodynamic quantities can be calculated using the formalism of the sum of states . The number of micro-states is counted for certain boundary conditions . The distribution of the microstates in phase space is given by the density of states .

Examples

Classic gas

Simulation of an ideal gas in two dimensions. Some molecules are drawn in red to make it easier to follow their movement

In classical physics, a gas is assumed to be a set of point-like particles of mass . The particles have the positions and the velocities . The index numbers the particles, the index is the number of a possible microstate; a microstate is the specification of the positions and current velocities of all particles at a certain point in time. ${\ displaystyle N}$ ${\ displaystyle m}$ ${\ displaystyle {\ vec {x}} _ {i, j}}$ ${\ displaystyle {\ vec {v}} _ {i, j}}$ ${\ displaystyle j}$ ${\ displaystyle i}$

The mean energy of the microstate can then be calculated from the kinetic energies of the gas particles:

{\ displaystyle E_ {i} \, = \, {\ frac {1} {N}} \ sum \ limits _ {j = 1} ^ {N} {\ frac {1} {2}} \, m \ , {\ vec {v}} _ {i, j} ^ {2}}

There are many states of energy , since only the velocities, but not the positions of the particles, contribute to this size. According to the postulate of the same a priori probabilities, each of these states has the same probability. ${\ displaystyle E_ {i}}$

The particles collide elastically against one another and against the walls of the vessel. As a result, after some time a thermal equilibrium is established in which the distribution of the velocities of the individual particles follows the Maxwell-Boltzmann distribution .

The probability for a state with the energy is: ${\ displaystyle E_ {i}}$

{\ displaystyle p_ {i} \, \ propto \, \ exp \ left (- {\ frac {E_ {i}} {k _ {\ mathrm {B}} \, T}} \ right)}

It is

${\ displaystyle \ textstyle T}$ the temperature of the gas
${\ displaystyle \ textstyle k _ {\ mathrm {B}} \, T}$ the thermal energy of the gas and
${\ displaystyle \ textstyle \ exp \ left (- {\ frac {E_ {i}} {k _ {\ mathrm {B}} \, T}} \ right)}$ the Boltzmann factor .

Ising model

Another example of statistical physics is the Ising model . In this one-dimensional system of spins , particles are arranged in a row. The spin of each particle points either upwards or downwards . If there is also an external magnetic field with the field strength , the energy of a microstate can be calculated as: ${\ displaystyle N}$ ${\ displaystyle s _ {\ uparrow} = + 1}$ ${\ displaystyle s _ {\ downarrow} = - 1}$ ${\ displaystyle H}$

{\ displaystyle E_ {i} \, = \, \ mu _ {\ mathrm {B}} \, H \, \ sum \ limits _ {j = 1} ^ {N} s_ {j}}

It is the Bohr magneton . In this case one can write down the possible micro-states and their energy directly: ${\ displaystyle \ mu _ {\ mathrm {B}}}$ ${\ displaystyle N = 3}$

Microstate	↑↑↑	↑↑ ↓	↑ ↓ ↑	↓ ↑↑	↑ ↓↓	↓ ↑ ↓	↓↓ ↑	↓↓↓
energy ${\ displaystyle E_ {i} / (\ mu _ {\ mathrm {B}} \, H)}$	3	1	1	1	-1	-1	-1	-3

In this example, too, a macrostate of a given energy can be represented by different microstates.

literature

Most statistical physics textbooks, such as:

Nolting : Basic Course Theoretical Physics 6: Statistical Physics . Springer DE, October 1, 2007, ISBN 978-3-540-68870-9 (accessed on January 6, 2013).
Schwabl : Statistical Mechanics . Springer-Verlag, Berlin, 3rd edition 2006, ISBN 978-3-540-31095-2
Wachter, Hoeber: Theoretical Physics . Springer publishing house. ISBN 3-540-21457-7

Individual evidence

↑ Wolfgang Nolting: Basic Course Theoretical Physics 6: Statistical Physics . Springer DE, October 1, 2007, ISBN 978-3-540-68870-9 , pp. 4–5 (accessed on January 6, 2013).

[Nolting2007-1] Wolfgang Nolting: Basic Course Theoretical Physics 6: Statistical Physics . Springer DE, October 1, 2007, ISBN 978-3-540-68870-9 , pp. 4–5 (accessed on January 6, 2013).