In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on the properties of small, finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of phase transitions in the thermodynamic limit. Lee–Yang theory constitutes an indispensable part of the theories of phase transitions. Originally developed for the Ising model, the theory has been extended and applied to a wide range of models and phenomena, including protein folding, percolation, complex networks, and molecular zippers. The theory is named after the Nobel laureates Tsung-Dao Lee and Yang Chen-Ning, who were awarded the 1957 Nobel Prize in Physics for their unrelated work on parity non-conservation in weak interaction.
Introduction For an equilibrium system in the canonical ensemble, all statistical information about the system is encoded in the partition function,
Z = ∑ i e − β E i , {\displaystyle Z=\sum _{i}e^{-\beta E_{i}},}
where the sum runs over all possible microstates, and β = 1 / ( k B T ) {\displaystyle \beta =1/(k_{B}T)} is the inverse temperature, k B {\displaystyle k_{B}} is the Boltzmann constant and E i {\displaystyle E_{i}} is the energy of a microstate. The moments ⟨ E n ⟩ {\displaystyle \langle E^{n}\rangle } of the energy statistics are obtained by differentiating the partition function with respect to the inverse temperature multiple times,
⟨ E n ⟩ = 1 Z ∂ − β n Z = ∑ i E i n e − β E i ∑ i e − β E i . {\displaystyle \langle E^{n}\rangle ={\frac {1}{Z}}\partial _{-\beta }^{n}Z={\frac {\sum _{i}E_{i}^{n}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}.}
From the partition function, we may also obtain the free energy
F = − β − 1 log [ Z ] . {\displaystyle F=-\beta ^{-1}\log[Z].}
Analogously to how the partition function generates the moments, the free energy generates the cumulants of the energy statistics
⟨ ⟨ E n ⟩ ⟩ = ∂ − β n ( − β F ) . {\displaystyle \langle \!\langle E^{n}\rangle \!\rangle =\partial _{-\beta }^{n}(-\beta F).}
More generally, if the microstate energies E i ( q ) = E i ( 0 ) − q Φ i {\displaystyle E_{i}(q)=E_{i}(0)-q\Phi _{i}} depend on a control parameter q {\displaystyle q} and a fluctuating conjugate variable Φ {\displaystyle \Phi } (whose value may depend on the microstate), the moments of Φ {\displaystyle \Phi } may be obtained as
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