In the
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
of
spin glass
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called 'freezing temperature' ''Tf''. In ferromagnetic solids, component atoms' magne ...
es and other systems with
quenched disorder
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.
In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one ...
, the replica trick is a mathematical technique based on the application of the formula:
or:
where
is most commonly the
partition function, or a similar thermodynamic function.
It is typically used to simplify the calculation of
, reducing the problem to calculating the disorder average
where
is assumed to be an integer. This is physically equivalent to averaging over
copies or ''replicas'' of the system, hence the name.
The crux of the replica trick is that while the disorder averaging is done assuming
to be an integer, to recover the disorder-averaged logarithm one must send
continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the same results. (To prove that the replica trick works, one would have to prove that
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not co ...
holds, that is, that the ratio
is of
exponential type
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function ''e'C'', ''z'', for some real-valued constant ''C'' as , ''z'', → ∞ ...
less than
pi.)
It is occasionally necessary to require the additional property of ''replica
symmetry breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
'' (RSB) in order to obtain physical results, which is associated with the breakdown of
ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
.
General formulation
It is generally used for computations involving
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s (can be expanded in power series).
Expand
using its
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
: into powers of
or in other words replicas of
, and perform the same computation which is to be done on
, using the powers of
.
A particular case which is of great use in physics is in averaging the
thermodynamic free energy
The thermodynamic free energy is a concept useful in the thermodynamics of chemical or thermal processes in engineering and science. The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process a ...
,
: