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Transport Phenomena

In-Gee Kim

Published Mar 14, 2018• 0 citations
Publisher

Abstract

So far we have studied many-body systems by considering the effect of adding or removing one or more particles. From the one particle Green’s function G ( 1 - 1 ′ ) , $ G(1 - 1'), $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math7_1.tif"/> we were able to determine the energy spectrum and decay times of the single-particle excited states.

DOI
10.1201/9781315196596-7

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Physical prerequisites

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The quantum revolution in the beginning of the 20th century changed our concepts of dynamics. The dynamic variables such as position r $ {\mathbf{r}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_1.tif"/> and momentum p $ {\mathbf{p}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_2.tif"/> are replaced by the corresponding position and momentum operators r ^ $ {\hat{\mathbf{r}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_3.tif"/> and p ^ $ {\hat{\mathbf{p}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_4.tif"/> , respectively When one would like to observe a dynamical variable, say ω , of a particle, one has to introduce a wavefunction ψ , which contains all the dynamical information of particles, and to apply the corresponding dynamical operator Ω ^ $ \hat{\Omega } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_5.tif"/> to the wavefunction ψ . Then one may obtain the desired dynamical value of the particle as the eigenvalue of the operator, 1.1 Ω ^ ψ = ω ψ . $$ \hat{\Omega }\psi = \omega \psi . $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math1_1.tif"/>

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The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .

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The nonequilibrium theory described in the previous chapters reduces to a particularly simple form for a system of fermions very close to zero temperature. To see this, let us define a “local occupation number” f ( p , ω ; R , T ) $ f(p, \omega ;R, T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math12_1.tif"/> by writing 12.1 g < ( p , ω ; R , T ) = a ( p , ω ; R , T ) f ( p , ω ; R , T ) $$ g^{ < } (p,~\omega ;R,~T) = a(p,~\omega ;R,~T)f(p,~\omega ;R,~T) $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math12_1.tif"/>

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Transport Equations for Many-Body Systems by Methods of Thermodynamic Green's Functions.

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So far we have studied many-body systems by considering the effect of adding or removing one or more particles. From the one particle Green’s function G ( 1 - 1 ′ ) , G(1−1′),G(1 - 1'),G(1−1′), https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math7_1.tif"/> we were able to determine the energy spectrum and decay times of the single-particle excited states.
The quantum revolution in the beginning of the 20th century changed our concepts of dynamics. The dynamic variables such as position r $ {\mathbf{r}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_1.tif"/> and momentum p $ {\mathbf{p}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_2.tif"/> are replaced by the corresponding position and momentum operators r ^ $ {\hat{\mathbf{r}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_3.tif"/> and p ^ $ {\hat{\mathbf{p}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_4.tif"/> , respectively When one would like to observe a dynamical variable, say ω , of a particle, one has to introduce a wavefunction ψ , which contains all the dynamical information of particles, and to apply the corresponding dynamical operator Ω ^ $ \hat{\Omega } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math1_5.tif"/> to the wavefunction ψ . Then one may obtain the desired dynamical value of the particle as the eigenvalue of the operator, 1.1 Ω ^ ψ = ω ψ . Ω^ψ=ωψ. \hat{\Omega }\psi = \omega \psi . Ω^ψ=ωψ. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math1_1.tif"/>
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .
The nonequilibrium theory described in the previous chapters reduces to a particularly simple form for a system of fermions very close to zero temperature. To see this, let us define a “local occupation number” f ( p , ω ; R , T ) $ f(p, \omega ;R, T) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math12_1.tif"/> by writing 12.1 g < ( p , ω ; R , T ) = a ( p , ω ; R , T ) f ( p , ω ; R , T ) g<(p, ω;R, T)=a(p, ω;R, T)f(p, ω;R, T) g^{ < } (p,~\omega ;R,~T) = a(p,~\omega ;R,~T)f(p,~\omega ;R,~T) g<(p, ω;R, T)=a(p, ω;R, T)f(p, ω;R, T) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math12_1.tif"/>