Transfer-matrix method
In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier.[1][2] In many one dimensional lattice models, the partition function is first written as an n-fold summation over each possible microstate, and also contains an additional summation of each component's contribution to the energy of the system within each microstate.
Overview
Higher-dimensional models contain even more summations. For systems with more than a few particles, such expressions can quickly become too complex to work out directly, even by computer.
Instead, the partition function can be rewritten in an equivalent way. The basic idea is to write the partition function in the form
where v0 and vN+1 are vectors of dimension p and the p × p matrices Wk are the so-called transfer matrices. In some cases, particularly for systems with periodic boundary conditions, the partition function may be written more simply as
where "tr" denotes the
The transfer-matrix method is used when the total system can be broken into a sequence of subsystems that interact only with adjacent subsystems. For example, a three-dimensional cubical lattice of
Importantly, transfer matrix methods allow to tackle probabilistic lattice models from an algebraic perspective, allowing for instance the use of results from representation theory.
As an example of observables that can be calculated from this method, the probability of a particular state occurring at position x is given by:
Where is the projection matrix for state , having elements
Transfer-matrix methods have been critical for many exact solutions of problems in
See also
References
Notes
- ISBN 978-0-12-083182-1.
- Teif V.B. (2007). "General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation". Nucleic Acids Res. 35 (11): e80. PMID 17526526.
- Efremov AK, Winardhi RS, Yan J (2016). "Transfer-matrix calculations of DNA polymer micromechanics under tension and torque constraints". Phys. Rev. E. 94 (3): 032404. PMID 27739846.
- Efremov AK, Yan J (2018). "Transfer-matrix calculations of the effects of tension and torque constraints on DNA-protein interactions". Nucleic Acids Res. 46 (13): 6504–6527. PMID 29878241.