[next] [prev] [prev-tail] [tail] [up]

3.6 Computational task

The computational task at hand is to calculate the statistical mean of an arbitrary physical quantity, represented by A (r,k), at time T using the Wigner function:

∫ ∫ ⟨AT ⟩ = dr dkfw (r,k, T)A (r,k). (3.25)
The mean of a physical quantity is often of greater interest than the Wigner function itself (one can solve for fw by setting A (r,k ) = δ (r)).

It is formally proven in [117] that a linear functional of the form (3.25) can be solved by calculating the statistical expectation value of a random variable, which is calculated using the procedure discussed in Section 3.5.

The RHS of (3.18) represents an inner product between gi and f, which is reminiscent of the inner product (3.25). Therefore, either f or g can be solved to obtain the mean value of a physical quantity. In the latter case the free term gi of the adjoint equation (3.17) is chosen to correspond to the physical quantity A, augmented with the time variable, such that:

g (Q) = A (r,k)δ (t - T) = A (Q) . (3.26) i T

The statistical mean of AT can be expressed as:

∫ ⟨AT ⟩ = dQg (Q )fi(Q) , (3.27)
which can be developed into a Neumann series of the form

∑∞ ⟨AT ⟩ = ⟨AT ⟩n . (3.28) n=0

The first expansion of g as a Fredholm equation, with the free term (3.26) yields:

∫ ∫ ∫ ∫ dQ1fi (Q1) g(Q1) = dQ1fi (Q1) AT (Q1)+ dQ1fi (Q1 ) dQK (Q, Q1 )AT (Q)+ ∫ ∫ ∫ (3.29) + dQ1fi (Q1) dQ dQ2K (Q, Q1)K (Q2, Q )g(Q2 ).
The first two terms of (3.28) are written here to convey the principle. The zero-th term simply contains the free term (3.26):
∫ ⟨AT ⟩0 = dQfi (Q)AT (Q ) ∫ ∞ ∫ ∫ ∫t1 = dt1 dk1 dr1fi(r1,k1)e- t0 μ(R1(y),k1)dyA (R1 (t1),k1 )δ(t1 - T ) ∫t0 ∫ - ∫tT0 μ(R1(y),k1)dy = dk1 dr1fi(r1,k1)e A (R1 (T ),k1). (3.30)
The meaning of the term is as follows: a particle initialized at (r ,k ,t ) 1 1 0 follows a trajectory R1 and reaches the point (R (T ),k ,T) 1 1 with a probability of e-∫ t0Tμ(R1(y),k1)dy and makes a contribution of A(R1 (T),k1) to the statistical mean of A. If the particle is scattered from its trajectory, it contributes through another term in the series.

∫ ∫ ⟨AT ⟩1 = dQ1fi (Q1 ) dQK (Q, Q1 )AT (Q) ∫ ∞ ∫ ∫ { - ∫t1μ(R1(y),k1)dy} = dt1 dk1 dr1fi(r1,k1) μ(R1 (t1),k1)e t0 ◟t0--------------------------◝◜----------------------------◞ ∫ ∫ { I } ∞ --Γ (r1,k,k1)- - ∫tt μ(R1(y),k)dy × θD(r1) t dt dk μ (R1 (t1) ,k1) e 1 A (R1 (t),k)δ (t - T). (3.31) ◟-------1------------------------◝◜---------------------------------◞ II
As before, a particle initialized at (r1,k1,t0) follows a trajectory R1, but only until time t1 where it is scattered. By extracting μ as a common factor, the term in the first set of braces can be interpreted as the probability for a particle to not be scattered until time t1 (the exponential) and then to be scattered in the interval dt1 thereafter (μ(R1 (t1),k1)dt1). The term in the braces of term II describes the probability of scattering from k1 to k and the exponential term again gives the probability that the scattered particle will not be scattered again until time T, where it contributes A(R1 (T),k ) to the statistical mean of A. If the particle is scattered again before time T, it contributes through another term in the series.

Further terms of the series can be written down in a similar manner to reveal the general structure: A particle that scatters n times before time T makes a contribution to ⟨AT ⟩ through the term ⟨AT⟩n in the series. The braced terms in the integrals of I and II represent probabilities, which clearly establishes the link to the Monte Carlo integration introduced in Section 3.5.

3.6.1 Wigner potential as a scattering mechanism

The term Γ (r ,k,k) μ(R11(t1),1k1)- appearing in term II represents the total scattering probability, including both phonons and the ’scattering’ (particle generation) associated to the Wigner potential. Phonon scattering occurs with a probability λ μ and is described in Section 3.7.5; the Wigner-related term is selected with a probability γ μ = ( ) 1 - λμ and can then be written as

{ V +(r,k′ - k ) V- (r,k′ - k) ( )} 3 1∕3-w------------ 1∕3-w-----------+ 1∕3δ k- k ′ . (3.32) γ (r) γ(r)
Each term is normalized by gamma to represent a probability, which follows from the definition
∫ + γ (r) ≡ dkVw (r,k) ≥ 0.

Two interpretations of (3.32) are possible: If considered as a scattering mechanism, one of the terms is selected, each with a probability 1∕3 and the particle is generated with a weight of ±3 at wavevector ±k′. Alternatively, all three terms can be chosen simultaneously and take a weight of ±1, i.e. two additional particles are created with wavevector ±k′ and weight ±1; the original particle, associated with the δ, persists unchanged. The algorithm to select the wavevectors of the generated particles is discussed in Section 3.7 and revisited in Chapter 4, which considers the semi-discrete form of the WBE.

[next] [prev] [prev-tail] [front] [up]