Binary and multiparticle contributions to time correlation functions

The virial coefficients for a quantum gas (including quantum statistics) are expressed as sums of cumulants of connected (generalized) Mayer diagrams, the cumulants being built on the irreducible blocks of the diagrams. The Mayer diagrams are defined for the quantum case in terms of imaginary time-ordered exponentials, the quantum statistics being incorporated in the guise of multiparticle interactions. In order to extend Mayer diagrams to multiparticle interactions, we utilize terminology and methods from the theory of hypergraphs. The virial coefficients naturally separate into a quantum Boltzmann gas contribution, an ideal quantum gas contribution, and a final term expressing correlations between dynamics and statistics. In the classical limit, connected Mayer diagrams factorize into their irreducible blocks; the cumulants over irreducible blocks then vanish (by a basic property of cumulants), except for diagrams which are themselves irreducible, whence the classical result of Mayer (extended to multiparticle interactions). In the quantum case, the imaginary time ordering prevents the factorization into irreducible blocks by time entangling them. As a further illustration of the use of hypergraph-cumulant methods, we directly deduce the expressions of the virial coefficients in terms of Ursell–Kahn–Uhlenbeck cluster functions (the ideal quantum gas contribution naturally appears in that form).The virial coefficients for a quantum gas (including quantum statistics) are expressed as sums of cumulants of connected (generalized) Mayer diagrams, the cumulants being built on the irreducible blocks of the diagrams. The Mayer diagrams are defined for the quantum case in terms of imaginary time-ordered exponentials, the quantum statistics being incorporated in the guise of multiparticle interactions. In order to extend Mayer diagrams to multiparticle interactions, we utilize terminology and methods from the theory of hypergraphs. The virial coefficients naturally separate into a quantum Boltzmann gas contribution, an ideal quantum gas contribution, and a final term expressing correlations between dynamics and statistics. In the classical limit, connected Mayer diagrams factorize into their irreducible blocks; the cumulants over irreducible blocks then vanish (by a basic property of cumulants), except for diagrams which are themselves irreducible, whence the classical result of Mayer (extended to multiparticle interactions). In the quantum case, the imaginary time ordering prevents the factorization into irreducible blocks by time entangling them. As a further illustration of the use of hypergraph-cumulant methods, we directly deduce the expressions of the virial coefficients in terms of Ursell–Kahn–Uhlenbeck cluster functions (the ideal quantum gas contribution naturally appears in that form).

Journal of Mathematical PhysicsApr 1, 19835 citations

Single-electron second-order correlation function G(2) at nonzero temperatures

Michael V. Moskalets

The single-particle state is not expected to demonstrate second-order coherence. Here I analyze the injection of electrons into a conductor and show that, at a nonzero temperature, the underlying Fermi sea causes the single-particle injected state to exhibit second-order coherence. For this purpose, I calculate the second-order correlation function, ${G}^{(2)}$ , of electrons injected on top of the Fermi sea. At zero temperature, the function ${G}^{(2)}$ unambiguously demonstrates whether the injected state is a single- or a multiparticle state: ${G}_{}^{(2)}$ vanishes in the former case, while it does not vanish in the latter case. However, at nonzero temperatures, when the injected state is a mixed state, a purely single-particle contribution makes the function ${G}_{}^{(2)}$ nonvanishing even in the case of injection of one electron. The single-particle contribution puts the lower limit on the second-order correlation function and thereby limits its use as a single-particle injection test at nonzero temperatures. The existence of a single-particle contribution to ${G}_{}^{(2)}$ can be verified experimentally by measuring the cross-correlation electrical noise.The single-particle state is not expected to demonstrate second-order coherence. Here I analyze the injection of electrons into a conductor and show that, at a nonzero temperature, the underlying Fermi sea causes the single-particle injected state to exhibit second-order coherence. For this purpose, I calculate the second-order correlation function, ${G}^{(2)}$, of electrons injected on top of the Fermi sea. At zero temperature, the function ${G}^{(2)}$ unambiguously demonstrates whether the injected state is a single- or a multiparticle state: ${G}_{}^{(2)}$ vanishes in the former case, while it does not vanish in the latter case. However, at nonzero temperatures, when the injected state is a mixed state, a purely single-particle contribution makes the function ${G}_{}^{(2)}$ nonvanishing even in the case of injection of one electron. The single-particle contribution puts the lower limit on the second-order correlation function and thereby limits its use as a single-particle injection test at nonzero temperatures. The existence of a single-particle contribution to ${G}_{}^{(2)}$ can be verified experimentally by measuring the cross-correlation electrical noise.

arXiv:1805.09860Physical review. B./Physical review. BSep 11, 20188 citations

Studies in Time-Dependent Correlation Functions.

G. Yue

Oct 1, 19760 citations

{G}^{(2)}

, of electrons injected on top of the Fermi sea. At zero temperature, the function

{G}^{(2)}

unambiguously demonstrates whether the injected state is a single- or a multiparticle state:

{G}_{}^{(2)}

vanishes in the former case, while it does not vanish in the latter case. However, at nonzero temperatures, when the injected state is a mixed state, a purely single-particle contribution makes the function

{G}_{}^{(2)}

nonvanishing even in the case of injection of one electron. The single-particle contribution puts the lower limit on the second-order correlation function and thereby limits its use as a single-particle injection test at nonzero temperatures. The existence of a single-particle contribution to

{G}_{}^{(2)}

can be verified experimentally by measuring the cross-correlation electrical noise.