Editing Quantum statistical mechanics

{{Short description|Statistical mechanics of quantum-mechanical systems}}
{{No footnotes|date=September 2024}}

{{Modern physics}}
{{Quantum mechanics|cTopic=Advanced topics}}

'''Quantum statistical mechanics''' is [[statistical mechanics]] applied to [[quantum mechanics|quantum mechanical systems]]. 

== Expectation ==
{{See also|Expectation value (quantum mechanics)|Density matrix#Measurement}}
In quantum mechanics a [[statistical ensemble (mathematical physics)|statistical ensemble]] ([[probability distribution]] over possible [[quantum state]]s) is described by a [[density matrix|density operator]] ''S'', which is a non-negative, [[self-adjoint]], [[trace-class]] operator of trace 1 on the [[Hilbert space]] ''H'' describing the quantum system.

From classical probability theory, we know that the [[expected value|expectation]] of a [[random variable]] ''X'' is defined by its [[Probability distribution|distribution]] D<sub>''X''</sub> by
<math display="block"> \mathbb{E}(X) = \int_\mathbb{R}d \lambda \operatorname{D}_X(\lambda) </math>
assuming, of course, that the random variable is [[integrable]] or that the random variable is non-negative. Similarly, let ''A'' be an [[observable]] of a quantum mechanical system. ''A'' is given by a [[densely defined]] [[self-adjoint operator]] on ''H''.  The [[spectral measure]] of ''A'' defined by

<math display="block"> \operatorname{E}_A(U) = \int_U d\lambda \operatorname{E}(\lambda), </math>

uniquely determines ''A'' and conversely, is uniquely determined by ''A''.  E<sub>''A''</sub>  is a [[Boolean homomorphism]] from the [[Borel subset]]s of '''R''' into the [[Lattice (order)|lattice]] ''Q'' of self-adjoint projections of ''H''. In analogy with probability theory, given a state ''S'', we introduce the ''distribution'' of ''A''  under ''S'' which is the probability measure defined on the Borel subsets of '''R''' by
<math display="block"> \operatorname{D}_A(U) = \operatorname{Tr}(\operatorname{E}_A(U) S). </math>
Similarly, the expected value of ''A'' is defined in terms of the probability distribution D<sub>''A''</sub> by
<math display="block"> \mathbb{E}(A) = \int_\mathbb{R} d\lambda \,  \operatorname{D}_A(\lambda).</math>
Note that this expectation is relative to the  mixed state ''S'' which is used in the definition of D<sub>''A''</sub>.

'''Remark'''.  For technical reasons, one needs to consider separately the positive and negative parts of ''A'' defined by the [[Borel functional calculus]] for unbounded operators.

One can easily show:
<math display="block"> \mathbb{E}(A)  = \operatorname{Tr}(A S) = \operatorname{Tr}(S A). </math>
The [[trace of an operator]] ''A'' is written as follows:
<math display="block"> \operatorname{Tr}(A) = \sum_{m} \langle m  |  A |  m \rangle  . </math> 
Note that if ''S'' is a [[pure state]] corresponding to the [[Euclidean vector|vector]] <math>\psi</math>, then:
<math display="block"> \mathbb{E}(A) = \langle \psi | A | \psi \rangle. </math>

== Von Neumann entropy ==<!-- This section is linked from [[Physical information]] -->
{{main|Von Neumann entropy}}

Of particular significance for describing randomness of a state is the von Neumann entropy of ''S'' ''formally'' defined by
<math display="block"> \operatorname{H}(S) = -\operatorname{Tr}(S \log_2 S). </math>  
Actually, the operator  ''S'' log<sub>2</sub> ''S'' is not necessarily trace-class. However, if ''S'' is a non-negative self-adjoint operator not of trace class we define Tr(''S'') = +&infin;.  Also note that any density operator ''S'' can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form
<math display="block"> \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 & \cdots \\ 0 & \lambda_2 & \cdots & 0 & \cdots\\ \vdots & \vdots & \ddots & \\ 0 & 0 & &  \lambda_n & \\ \vdots & \vdots & & & \ddots \end{bmatrix} </math>
and we define
<math display="block"> \operatorname{H}(S) = - \sum_i \lambda_i \log_2 \lambda_i. </math>
The convention is that <math> \; 0 \log_2 0 = 0</math>, since an event with probability zero should not contribute to the entropy. This value is an extended real number (that is in [0, &infin;]) and this is clearly a unitary invariant of ''S''.

'''Remark'''. It is indeed possible that H(''S'') = +&infin; for some density operator ''S''. In fact ''T'' be the diagonal matrix
<math display="block"> T = \begin{bmatrix} \frac{1}{2 (\log_2  2)^2 }& 0 & \cdots & 0 & \cdots \\ 0 & \frac{1}{3 (\log_2  3)^2 } & \cdots & 0 & \cdots\\ \vdots & \vdots & \ddots &  \\ 0 & 0 & &  \frac{1}{n (\log_2  n)^2 } & \\ \vdots & \vdots & & & \ddots \end{bmatrix} </math>
''T'' is non-negative trace class and one can show ''T'' log<sub>2</sub> ''T'' is not trace-class.

{{math theorem | Entropy is a unitary invariant.}}

In analogy with [[Shannon entropy#Formal definitions|classical entropy]] (notice the similarity in the definitions), H(''S'') measures the amount of randomness in the state ''S''. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space ''H'' is finite-dimensional, entropy is maximized for the states ''S'' which in diagonal form have the representation
<math display="block"> \begin{bmatrix} \frac{1}{n} & 0 & \cdots & 0 \\ 0 & \frac{1}{n} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots &  \frac{1}{n} \end{bmatrix} </math>
For such an ''S'', H(''S'') = log<sub>2</sub> ''n''. The state ''S'' is called the maximally mixed state.

Recall that a [[pure state]] is one of the form 
<math display="block"> S = | \psi \rangle \langle \psi |, </math>
for &psi; a vector of norm 1.

{{math theorem | math_statement =  {{math|1=H(''S'') = 0}} if and only if {{mvar|S}} is a pure state.}}

For {{mvar|S}} is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1.

Entropy can be used as a measure of [[quantum entanglement]].

== Gibbs canonical ensemble ==

{{main|canonical ensemble}}

Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''.  If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +&infin; sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.

The ''[[Gibbs canonical ensemble]]'' is described by the state 
<math display="block"> S= \frac{\mathrm{e}^{- \beta H}}{\operatorname{Tr}(\mathrm{e}^{- \beta H})}. </math>
Where &beta; is such that the ensemble average of energy satisfies  
<math display="block"> \operatorname{Tr}(S H) = E </math>

and

<math display="block">\operatorname{Tr}(\mathrm{e}^{- \beta H}) = \sum_n \mathrm{e}^{- \beta E_n} = Z(\beta) </math>

This is called the [[partition function (mathematics)|partition function]]; it is the quantum mechanical version of the [[canonical partition function]] of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue <math>E_m</math> is

<math display="block">\mathcal{P}(E_m) = \frac{\mathrm{e}^{- \beta E_m}}{\sum_n \mathrm{e}^{- \beta E_n}}.</math>

The Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the condition that the average energy is fixed.

== Grand canonical ensemble ==

{{main|grand canonical ensemble}}

For open systems where the energy and numbers of particles may fluctuate, the system is described by the [[grand canonical ensemble]], described by the density matrix
<math display="block"> \rho = \frac{\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}}{\operatorname{Tr}\left(\mathrm{e}^{ \beta ( \sum_i \mu_iN_i - H)}\right)}. </math>
where the ''N''<sub>1</sub>, ''N''<sub>2</sub>, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Note that this is a density matrix including many more states (of varying N) compared to the canonical ensemble.

The grand partition function is
<math display="block">\mathcal Z(\beta, \mu_1, \mu_2, \cdots) = \operatorname{Tr}(\mathrm{e}^{\beta (\sum_i \mu_iN_i - H)}) </math>

==See also==
* [[Quantum thermodynamics]]
* [[Thermal quantum field theory]]
* [[Stochastic thermodynamics]]
* [[Abstract Wiener space]]

== Further reading ==
{{reflist}}
{{refbegin}}
* Modern review for closed systems: {{Cite journal |last=Nandkishore |first=Rahul |last2=Huse |first2=David A. |date=2015-03-10 |title=Many-Body Localization and Thermalization in Quantum Statistical Mechanics |url=https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031214-014726 |journal=Annual Review of Condensed Matter Physics |language=en |volume=6 |pages=15–38 |doi=10.1146/annurev-conmatphys-031214-014726 |issn=1947-5454}}
* {{Cite book |last=Schieve |first=William C. |title=Quantum statistical mechanics |date=2009 |publisher=Cambridge University Press |isbn=978-0-521-84146-7 |location=Cambridge, UK}} 
* Field theory methods applied to quantum many body problems. {{Cite book |last=Kadanoff |first=Leo P. |url=https://www.taylorfrancis.com/books/9780429961762 |title=Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Nonequilibrium Problems |last2=Baym |first2=Gordon |date=2018-03-08 |publisher=CRC Press |isbn=978-0-429-49321-8 |edition=1 |language=en |doi=10.1201/9780429493218}} 
* Advanced graduate textbook {{Cite book |last=Bogoli︠u︡bov |first=N. N. |url=https://www.worldcat.org/title/526687587 |title=Introduction to quantum statistical mechanics |last2=Bogoli︠u︡bov |first2=N. N. |date=2010 |publisher=World Scientific |isbn=978-981-4295-19-2 |edition=2 |location=Hackensack, NJ |oclc=526687587}}
{{refend}}

{{Quantum mechanics topics}}

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[[Category:Quantum mechanical entropy]]