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The Bethe ansatz solution for particles jumping on a infinite lattice was given in. Our solution for the infinite system, Eqs. d12 d13 d14 and φ (kj, xj,0) = e-i kj xj,0, can be obtained from the solution in by the replacements ∫0 2π d kj /2π → - d kj /2π and by expanding the energy E (k1,..., kN) and the scattering coefficients Sij in a power series in momenta to lowest order.
-
The Bethe ansatz solution for particles jumping on a infinite lattice was given in. Our solution for the infinite system, Eqs. d12 d13 d14 and φ (kj, xj,0) = e-i kj xj,0, can be obtained from the solution in by the replacements ∫0 2π d kj /2π → - d kj /2π and by expanding the energy E (k1,..., kN) and the scattering coefficients Sij in a power series in momenta to lowest order.
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70449664456
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The Bethe ansatz solution is often written in terms of discrete wave vectors. Using the Poisson summation formula gives the identity m=- e2im kj = (π/' m=-δ (kj +mπ/ℓ), from which the Bethe ansatz solution, Eqs. d12 d13 d14 d15, can be rewritten in term of discrete wave vectors mj π/ℓ (mj =-∞,...,-1,0,1,...,∞) if desired.
-
The Bethe ansatz solution is often written in terms of discrete wave vectors. Using the Poisson summation formula gives the identity m=- e2im kj = (π/' m=-δ (kj +mπ/ℓ), from which the Bethe ansatz solution, Eqs. d12 d13 d14 d15, can be rewritten in term of discrete wave vectors mj π/ℓ (mj =-∞,...,-1,0,1,...,∞) if desired.
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Note that our convention, with coordinates in the range [-L/2,L/2], is different to Ref. where the left (right) end of the system is at 0 (L).
-
Note that our convention, with coordinates in the range [-L/2,L/2], is different to Ref. where the left (right) end of the system is at 0 (L).
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47
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70449676814
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Alternatively, one may be interested in the tPDF for a fixed initial particle distribution x 0: ρ T (xT, t x 0) = R d x1′ d xN′ δ (xT - xT′) P (x ′, t x0). For the case where all particles start at the same position a straightforward integration yields, using Eq., a result for ρ T (xT, t x 0) which in the limit → is identical to that obtained in, as it should.
-
Alternatively, one may be interested in the tPDF for a fixed initial particle distribution x 0: ρ T (xT, t x 0) = R d x1′ d xN′ δ (xT - xT′) P (x ′, t x 0). For the case where all particles start at the same position a straightforward integration yields, using Eq., a result for ρ T (xT, t x 0) which in the limit → is identical to that obtained in, as it should.
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The expression for the tPDF, Eq. is (for Δ=0) related to Eq. (61) in via elementary recursion relations for the Jacobi polynomial.
-
The expression for the tPDF, Eq., is (for Δ=0) related to Eq. (61) in via elementary recursion relations for the Jacobi polynomial.
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50
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70449656273
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The variable ξ defined in Eq. is related to the variable y in Ref. according to ξ=1/y.
-
The variable ξ defined in Eq. is related to the variable y in Ref. according to ξ=1/y.
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We inverted the definition (see Ref.) cosh2ζ= (1+ξ) / (1-ξ) to arrive at Eq. d33.
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We inverted the definition (see Ref.) cosh2ζ= (1+ξ) / (1-ξ) to arrive at Eq. d33.
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In more detail, in arriving at the SFD expression for ρT (yT, t yT,0) we assumed 1/N1, δN /N1, (4Dt) / (ℓ/2± xT) 2 1, (4Dt) / (ℓ/2± xT,0) 2 1, |η| 1, xT,0 /1, and | δN (xT - xT,0) η| /1.
-
In more detail, in arriving at the SFD expression for ρT (yT, t yT,0) we assumed 1/N1, δN /N1, (4Dt) / (ℓ/2± xT) 2 1, (4Dt) / (ℓ/2± xT,0) 2 1, |η| 1, xT,0 /1, and | δN (xT - xT,0) η| /1.
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The result in Eq. d43 (large N) can also be obtain from Eq. d35. Using Iα (z) z1 ez / 2πz (but not assuming that ψLR,ψRL are small like in Sec. s5B) one shows that the large- N long-time result (ξ→1), as determined by Eq. d35, agrees with Eq. d43.
-
The result in Eq. d43 (large N) can also be obtain from Eq. d35. Using Iα (z) z1 ez / 2πz (but not assuming that ψLR,ψRL are small like in Sec. s5B) one shows that the large- N long-time result (ξ→1), as determined by Eq. d35, agrees with Eq. d43.
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