Micro-canonical ensemble

In statistical physics, the micro-canonical ensemble describes a system with a fixed total energy in thermodynamic equilibrium . It differs from the canonical ensemble , because this describes a system whose energy can fluctuate because it is in thermal contact with an environment with a fixed temperature.

The only information about a quantum system is that the total energy is the same or out of the interval with , whereby the states must be compatible with externally specified parameters such as volume and number of particles. This corresponds to a system within a completely closed box (no energy or particle exchange with the environment, no external fields). The potential outside the box is infinite, so the Hamilton operator only has discrete energy eigenvalues ​​and the states can be counted. ${\displaystyle E=\langle {\hat {H}}\rangle }$ ${\displaystyle E_{0}}$ ${\displaystyle [E_{0},E_{0}+\Delta E]}$ ${\displaystyle \Delta E\ll E}$

Quantum mechanics

In equilibrium, the system's density operator does not change . According to the Von Neumann equation , in equilibrium, the density operator interchanges with the Hamilton operator ( commutator equal to zero). Therefore common eigen-states can be chosen, i. H. the energy eigenstates of are also eigenstates of . ${\displaystyle {\tfrac {\partial {\hat {\rho }}}{\partial t}}=0}$ ${\displaystyle {\tfrac {\partial {\hat {\rho }}}{\partial t}}=-{\tfrac {i}{\hbar }}[{\hat {H}},{\hat {\rho }}]}$ ${\displaystyle {\hat {H}}}$ ${\displaystyle {\hat {\rho }}}$

The Hilbert space is restricted to a subspace that is spanned by the state vectors with eigenvalues (eigenspace). Let be a complete orthonormal system (VONS), namely the eigen-states of the Hamilton operator , then the subspace is spanned by the basis vectors for which . The energy is generally degenerate (that is why a state is not uniquely determined by specifying , but a further quantum number is necessary); the degree of degeneracy corresponds to the dimension of the subspace . ${\displaystyle {\mathcal {H}}}$ ${\displaystyle E_{0}}$ ${\displaystyle \left\{|E,n\rangle \right\}}$ ${\displaystyle {\hat {H}}|E,n\rangle =E|E,n\rangle }$ ${\displaystyle {\mathcal {H}}_{E_{0}}}$ ${\displaystyle |E,n\rangle }$ ${\displaystyle E=E_{0}}$ ${\displaystyle E_{0}}$ ${\displaystyle E}$ ${\displaystyle n}$ ${\displaystyle g}$ ${\displaystyle \operatorname {dim} {\mathcal {H}}_{E_{0}}}$

The Hamilton operator cannot distinguish between the base states ( degeneracy ); Since no base state is preferred, the same probability is assigned a priori to each base state : According to the maximum entropy method , the system is to be described by the state which maximizes entropy . The entropy becomes maximal if and only if every basis vector has the same probability . ${\displaystyle {\hat {H}}}$ ${\displaystyle g}$ ${\displaystyle E_{0}}$ ${\displaystyle {\hat {H}}}$ ${\displaystyle 1/g}$ ${\displaystyle 1/\operatorname {dim} {\mathcal {H}}_{E_{0}}}$

Hence the density operator of the micro-canonical ensemble results in

${\displaystyle {\hat {\rho }}_{E_{0}}={\frac {1}{Z_{m}(E_{0})}}\sum _{n=1}^{\operatorname {dim} {\mathcal {H}}_{E_{0}}}|E_{0},n\rangle \langle E_{0},n|}$

with the micro-canonical partition function (often also referred to as) ${\displaystyle \Omega }$

${\displaystyle Z_{m}(E_{0})=\operatorname {dim} {\mathcal {H}}_{E_{0}}=\operatorname {Tr} \left(\sum _{n=1}^{{\textrm {dim}}{\mathcal {H}}_{E_{0}}}|E_{0},n\rangle \langle E_{0},n|\right),}$

where the trace of an operator is defined as follows: for any FROM of ${\displaystyle \operatorname {Tr} (O)=\sum _{m}\langle m|O|m\rangle }$ ${\displaystyle \left\{|m\rangle \right\}}$ ${\displaystyle {\mathcal {H}}}$

The basic assumption of equilibrium statistics is that every energy eigenstate with energy has the same probability. All equilibrium properties for closed or open systems can be derived from it (e.g. canonical or grand canonical ensemble). ${\displaystyle E_{0}}$

Classic

The classical microcanonical state of equilibrium for particles (phase space density) results analogously ${\displaystyle N}$

${\displaystyle \rho _{E_{0}}({\vec {x}}_{1},{\vec {p}}_{1},\ldots ,{\vec {x}}_{N},{\vec {p}}_{N})={\frac {1}{Z_{m}(E_{0})}}\delta (E_{0}-H({\vec {q}}_{1},{\vec {p}}_{1},\ldots ,{\vec {q}}_{N},{\vec {p}}_{N}))}$

with the classical microcanonical partition function (total number of accessible microstates that have the same total energy ) ${\displaystyle E_{0}}$

${\displaystyle Z_{m}(E_{0})=\int _{\mathbb {R} ^{6N}}\mathrm {d} \tau \;\delta (E_{0}-H({\vec {q}}_{1},{\vec {p}}_{1},\ldots ,{\vec {q}}_{N},{\vec {p}}_{N}))}$

With ${\displaystyle \mathrm {d} \tau ={\frac {1}{\xi h^{3N}}}\;\mathrm {d} ^{3}q_{1}\mathrm {d} ^{3}p_{1}\ldots \mathrm {d} ^{3}q_{N}\mathrm {d} ^{3}p_{N}}$

where for N identical particles the factor prevents the multiple counting of indistinguishable particles , ${\displaystyle \xi =N!}$

and for different types of particles with particle numbers and the factor . ${\displaystyle n}$ ${\displaystyle N_{1},\ldots ,N_{n}}$ ${\displaystyle \sum _{i=1}^{n}N_{i}=N}$ ${\displaystyle \xi =N_{1}!\ldots N_{n}!}$

The micro-canonical sum of states can be understood as the surface of the energy hypersurface and is therefore the derivation of the volume of the energy shell : ${\displaystyle Z_{m}}$ ${\displaystyle H-E=0}$ ${\displaystyle \Gamma _{m}(E)}$

${\displaystyle \Gamma _{m}(E)=\int _{\mathbb {R} ^{6N}}\mathrm {d} \tau \;\Theta (E-H)\quad \implies \quad Z_{m}={\frac {\mathrm {d} \Gamma (E)}{\mathrm {d} E}}}$,

where represents the Heaviside function . ${\displaystyle \Theta }$

Leads to coordinates on the -dimensional energy cup and a coordinate that is perpendicular thereto, and decomposes the Hamilton function with , then the partition function can be written as a surface integral: ${\displaystyle \tau _{E}}$ ${\displaystyle (6N-1)}$ ${\displaystyle \tau _{\perp }}$ ${\displaystyle H(q,p)=H(\tau _{E})+|\nabla H|\,\tau _{\perp }}$ ${\displaystyle H(\tau _{E})=E}$

${\displaystyle Z_{m}(E)=\int _{\mathbb {R} ^{6N-1}}\mathrm {d} \tau _{E}\int _{\mathbb {R} }\mathrm {d} \tau _{\perp }\;\underbrace {\delta \!\left(E-H(\tau _{E})-|\nabla H|\,\tau _{\perp }\right)} _{|\nabla H|^{-1}\delta (\tau _{\perp })}=\int _{\mathbb {R} ^{6N-1}}\mathrm {d} \tau _{E}{\frac {1}{|\nabla H|}}}$.

The gradient of the Hamilton function is perpendicular to the speed in phase space , so that the speed is always tangential to the energy shell. But both are identical in terms of magnitude: . Thus, the sum of the surface elements is divided by the speed in the phase space, so that areas with high speed contribute less to the integral (see also: ergodic hypothesis ). ${\displaystyle \nabla H=({\tfrac {\partial H}{\partial q}},{\tfrac {\partial H}{\partial p}})}$ ${\displaystyle {\boldsymbol {v}}=({\dot {q}},{\dot {p}})=({\tfrac {\partial H}{\partial p}},-{\tfrac {\partial H}{\partial q}})}$ ${\displaystyle {\boldsymbol {v}}\cdot \nabla H=0}$ ${\displaystyle |{\boldsymbol {v}}|=|\nabla H|}$ ${\displaystyle Z_{m}}$

entropy

The entropy can be expressed by the partition function:

${\displaystyle S({\hat {\rho }}_{E_{0}})=k_{\mathrm {B} }\ln \left[Z_{m}(E_{0})\right]}$

This can be derived from the definition of entropy, where the partition function is the same . ${\displaystyle Z_{m}(E_{0})=\operatorname {dim} {\mathcal {H}}_{E_{0}}=g}$

${\displaystyle {\begin{array}{rcl}S({\hat {\rho }}_{E_{0}})&=&-k_{\mathrm {B} }\left\langle \ln {\hat {\rho }}_{E_{0}}\right\rangle =-k_{\mathrm {B} }{\textrm {Tr}}\left[{\hat {\rho }}_{E_{0}}\ln {\hat {\rho }}_{E_{0}}\right]=-k_{\mathrm {B} }\sum _{E,m}\langle E,m|{\hat {\rho }}_{E_{0}}\ln {\hat {\rho }}_{E_{0}}|E,m\rangle \\&=&-k_{\mathrm {B} }\sum _{E,m}\langle E,m|\sum _{n=1}^{g}|E_{0},n\rangle {\frac {1}{g}}\langle E_{0},n|\ln \left(\sum _{n'=1}^{g}|E_{0},n'\rangle {\frac {1}{g}}\langle E_{0},n'|\right)|E,m\rangle \\&=&-k_{\mathrm {B} }\sum _{n=1}^{g}{\frac {1}{g}}\ln({\frac {1}{g}})=-k_{\mathrm {B} }\ln({\frac {1}{g}})=k_{\mathrm {B} }\ln \left(g\right)\end{array}}}$

thermodynamics

Entropy is the total energy (hereinafter referred to), the volume and number of particles dependent (because the Hamiltonian of and depends) . The total energy is usually called internal energy in thermodynamics . The derivatives of entropy are now examined. ${\displaystyle E_{0}}$ ${\displaystyle E}$ ${\displaystyle V}$ ${\displaystyle N}$ ${\displaystyle V}$ ${\displaystyle N}$ ${\displaystyle S(E,V,N)}$

temperature

If two closed systems in poor thermal contact, then only a little energy can be exchanged per unit of time, so as to remain in balance and can be written additively, the entropy of the entire system, the subsystems: . The total energy always remains constant . Increases the energy of a system, it must remove the other of the same extent: . The equilibrium of the entire system is achieved through the exchange of energy, where the total entropy is at a maximum: ${\displaystyle S_{\text{ges}}=S_{1}+S_{2}}$ ${\displaystyle E_{\text{ges}}=E_{1}+E_{2}}$ ${\displaystyle \mathrm {d} E_{1}=-\mathrm {d} E_{2}}$

${\displaystyle 0=\mathrm {d} S_{\text{ges}}={\frac {\partial S_{1}}{\partial E_{1}}}\mathrm {d} E_{1}+{\frac {\partial S_{2}}{\partial E_{2}}}\mathrm {d} E_{2}=\left({\frac {\partial S_{1}}{\partial E_{1}}}-{\frac {\partial S_{2}}{\partial E_{2}}}\right)\mathrm {d} E_{1}\quad \Rightarrow \quad {\frac {\partial S_{1}}{\partial E_{1}}}={\frac {\partial S_{2}}{\partial E_{2}}}}$

In equilibrium, the derivatives of the entropy according to the energy are the same for the subsystems. This defines the temperature :

${\displaystyle {\frac {1}{T}}:=\left({\frac {\partial S}{\partial E}}\right)_{N,V}\quad \Rightarrow \quad T:=\left({\frac {\partial E}{\partial S}}\right)_{N,V}}$

Thus, in equilibrium, the temperatures of both subsystems are the same.

pressure

The pressure is defined by

${\displaystyle p:=-\left\langle {\frac {\partial {\hat {H}}}{\partial V}}\right\rangle =-\left({\frac {\partial E}{\partial V}}\right)_{S,N}}$

i.e. the isentropic ( = const) change in energy per volume. The entropy is differentiated according to the volume: ${\displaystyle S}$ ${\displaystyle S(E(V,N),V,N)}$

${\displaystyle 0={\frac {\mathrm {d} S}{\mathrm {d} V}}=\left({\frac {\partial E}{\partial V}}\right)_{S,N}\left({\frac {\partial S}{\partial E}}\right)_{N,V}+\left({\frac {\partial S}{\partial V}}\right)_{E,N}=-p{\frac {1}{T}}+\left({\frac {\partial S}{\partial V}}\right)_{E,N}}$

Thus one obtains

${\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{E,N}={\frac {p}{T}}}$

Chemical potential

The chemical potential is defined by

${\displaystyle \mu :=\left\langle {\frac {\partial {\hat {H}}}{\partial N}}\right\rangle =\left({\frac {\partial E}{\partial N}}\right)_{S,V}}$

thus the isentropic change in energy per particle. The entropy is differentiated according to the number of particles: ${\displaystyle S(E(V,N),V,N)}$

${\displaystyle 0={\frac {\mathrm {d} S}{\mathrm {d} N}}=\left({\frac {\partial E}{\partial N}}\right)_{S,V}\left({\frac {\partial S}{\partial E}}\right)_{N,V}+\left({\frac {\partial S}{\partial N}}\right)_{E,V}=\mu {\frac {1}{T}}+\left({\frac {\partial S}{\partial N}}\right)_{E,V}}$

It follows

${\displaystyle \left({\frac {\partial S}{\partial N}}\right)_{E,V}=-{\frac {\mu }{T}}}$

In general it can be stated: If the Hamilton operator depends on an external parameter (e.g. volume or number of particles), then the derivative of the entropy is the same for constant energy: ${\displaystyle a}$ ${\displaystyle a}$

${\displaystyle \left({\frac {\partial S}{\partial a}}\right)_{E}=-{\frac {1}{T}}\left\langle {\frac {\partial {\hat {H}}}{\partial a}}\right\rangle }$

Thermodynamic potential

In summary, the derivatives according to energy, volume and number of particles can be represented:

${\displaystyle {\frac {1}{T}}{\begin{pmatrix}1\\p\\-\mu \end{pmatrix}}={\begin{pmatrix}\partial _{E}\\\partial _{V}\\\partial _{N}\end{pmatrix}}S(E,V,N)}$

The total differential of entropy is:

${\displaystyle \mathrm {d} S={\frac {1}{T}}\,\mathrm {d} E+{\frac {p}{T}}\,\mathrm {d} V-{\frac {\mu }{T}}\,\mathrm {d} N}$

The entropy can be resolved according to the energy . The energy is the thermodynamic potential of microcanonics. With it, the above derivatives can be written compactly as a gradient of the potential: ${\displaystyle S(E,V,N)}$ ${\displaystyle E(S,V,N)}$

${\displaystyle {\begin{pmatrix}T\\-p\\\mu \end{pmatrix}}={\begin{pmatrix}\partial _{S}\\\partial _{V}\\\partial _{N}\end{pmatrix}}E(S,V,N)}$

The total differential of the energy is thus:

${\displaystyle \mathrm {d} E=T\,\mathrm {d} S-p\,\mathrm {d} V+\mu \,\mathrm {d} N}$

This is the fundamental equation of thermodynamics.

Examples

Ideal gas

An example of a microcanonically prepared system that can be calculated with the classical methods is the ideal gas ; Derivation using the Sackur-Tetrode equation .

Uncoupled oscillators

Another example is a system of non-interacting harmonic oscillators of the same type . Whose total Hamilton operator is ${\displaystyle N}$

${\displaystyle {\hat {H}}=\sum _{i=1}^{N}\hbar \omega \left({\hat {n}}_{i}+{\frac {1}{2}}\right)}$

where is the population number operator of the -th oscillator. For a given total energy ${\displaystyle {\hat {n}}_{i}}$ ${\displaystyle i}$

${\displaystyle E=\sum _{i=1}^{N}\hbar \omega \left(n_{i}+{\frac {1}{2}}\right)=\hbar \omega \left(\sum _{i=1}^{N}n_{i}+{\frac {N}{2}}\right)\quad \Rightarrow \quad n:=\sum _{i=1}^{N}n_{i}={\frac {E}{\hbar \omega }}-{\frac {N}{2}}}$

the sum of states should now be calculated. This is equal to the degree of degeneracy to the energy E or the number of possibilities to distribute indistinguishable energy quanta to multiple occupant oscillators or the number of possibilities to distribute indistinguishable energy quanta to easily occupied oscillators (the combinatorial problem of distributing indistinguishable spheres to multiple occupant pots is equivalent to the task of arranging indistinguishable spheres and indistinguishable inner walls in a row): ${\displaystyle n}$ ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle N+n-1}$ ${\displaystyle n}$ ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle (N-1)}$

${\displaystyle Z_{m}={\binom {N+n-1}{n}}={\frac {(N+n-1)!}{n!\,(N-1)!}}}$

From this the entropy can be calculated:

${\displaystyle S=k_{\mathrm {B} }\ln(Z_{m})=k_{\mathrm {B} }\left\{\ln \left((N+n-1)!\right)-\ln(n!)-\ln \left((N-1)!\right)\right\}}$

For large and one can expand the logarithm of the faculty with the Stirling formula up to the first order and also neglect the 1 compared to the huge number : ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle \ln(N!)\approx N\ln(N)-N}$ ${\displaystyle N}$

${\displaystyle {\begin{aligned}S&=k_{\mathrm {B} }\left\{(N+n-1)\,\ln(N+n-1)-n\,\ln(n)-(N-1)\,\ln(N-1)\right\}\\&=k_{\mathrm {B} }\left\{(N+n)\,\ln(N+n)-n\,\ln(n)-N\,\ln(N)\right\}\end{aligned}}}$

Sorting and using of yields the entropy, which is extensive because of the prefactor ( and is namely intensive ): ${\displaystyle \epsilon :={\frac {E}{N\hbar \omega }}={\frac {n}{N}}+{\frac {1}{2}}}$ ${\displaystyle N}$ ${\displaystyle n/N}$ ${\displaystyle \epsilon }$

${\displaystyle {\begin{aligned}S&=Nk_{\mathrm {B} }\left\{\left({\frac {n}{N}}+1\right)\,\ln \left({\frac {n}{N}}+1\right)-\left({\frac {n}{N}}\right)\,\ln \left({\frac {n}{N}}\right)\right\}\\&=Nk_{\mathrm {B} }\left\{\left(\epsilon +{\frac {1}{2}}\right)\,\ln \left(\epsilon +{\frac {1}{2}}\right)-\left(\epsilon -{\frac {1}{2}}\right)\,\ln \left(\epsilon -{\frac {1}{2}}\right)\right\}\end{aligned}}}$

Now the temperature can be calculated:

${\displaystyle {\frac {1}{T}}={\frac {\partial S}{\partial E}}={\frac {\partial \epsilon }{\partial E}}{\frac {\partial S}{\partial \epsilon }}={\frac {k_{\mathrm {B} }}{\hbar \omega }}\ln {\frac {\epsilon +{\frac {1}{2}}}{\epsilon -{\frac {1}{2}}}}={\frac {k_{\mathrm {B} }}{\hbar \omega }}\ln {\frac {E+{\frac {N\hbar \omega }{2}}}{E-{\frac {N\hbar \omega }{2}}}}>0}$

For (the minimum total energy) is the temperature and increases strictly monotonically with the energy. For large , the temperature goes asymptotically towards . ${\displaystyle E\to N{\tfrac {\hbar \omega }{2}}}$ ${\displaystyle T=0}$ ${\displaystyle E}$ ${\displaystyle T={\tfrac {E}{Nk_{\mathrm {B} }}}+{\mathcal {O}}(E^{-1})}$

Finally one can still dissolve after the energy. The energy increases monotonically with the temperature:

${\displaystyle E=N\hbar \omega \left\{{\frac {1}{2}}+{\frac {1}{\exp({\frac {\hbar \omega }{k_{\mathrm {B} }T}})-1}}\right\}\geq N{\frac {\hbar \omega }{2}}}$

For is the total energy and increases strictly monotonically with temperature. For large ones the energy goes asymptotically against . ${\displaystyle T\to 0}$ ${\displaystyle E=N{\tfrac {\hbar \omega }{2}}}$ ${\displaystyle T}$ ${\displaystyle E=Nk_{\mathrm {B} }T+{\mathcal {O}}(T^{-1})}$

The chemical potential is:

${\displaystyle {\begin{aligned}\mu &=-T{\frac {\partial S}{\partial N}}=-{\frac {k_{\mathrm {B} }T}{2}}\,\ln \!\left[\left(\epsilon +{\frac {1}{2}}\right)\left(\epsilon -{\frac {1}{2}}\right)\right]\\&=-{\frac {\hbar \omega }{2}}\,{\frac {\ln \!\left(\epsilon +{\frac {1}{2}}\right)+\ln \!\left(\epsilon -{\frac {1}{2}}\right)}{\ln \!\left(\epsilon +{\frac {1}{2}}\right)-\ln \!\left(\epsilon -{\frac {1}{2}}\right)}}=k_{\mathrm {B} }T\,\ln \left[\exp \!\left({\frac {\hbar \omega }{k_{\mathrm {B} }T}}\right)-1\right]-{\frac {\hbar \omega }{2}}\leq {\frac {\hbar \omega }{2}}\end{aligned}}}$

For or is the chemical potential and falls strictly monotonically with the energy or with the temperature. For large or , the chemical potential becomes negative and goes asymptotically towards or . ${\displaystyle E\to N{\tfrac {\hbar \omega }{2}}}$ ${\displaystyle T\to 0}$ ${\displaystyle \mu ={\tfrac {\hbar \omega }{2}}}$ ${\displaystyle E}$ ${\displaystyle T}$ ${\displaystyle \mu =-{\tfrac {E}{N}}\ln {\tfrac {E}{N\hbar \omega }}+{\mathcal {O}}(E^{-1})}$ ${\displaystyle \mu =-k_{\mathrm {B} }T\ln {\tfrac {k_{\mathrm {B} }T}{\hbar \omega }}+{\mathcal {O}}(T^{-1})}$

literature

Wikibooks: Statistische_Mechanik / _Mikrokanonische_Ensemble  - learning and teaching materials

• Balian: From Microphysics to Macrophysics 1 . Springer-Verlag, Berlin, 2nd edition 2006, ISBN 3-540-45469-1
• Schwabl : Statistical Mechanics . Springer-Verlag, Berlin, 3rd edition 2006, ISBN 978-3-540-31095-2