Abstract
Thermodynamics is independent of a description at a microscopic level consequently statistical thermodynamics must produce results independent of the coordinate system used to describe the particles and their interactions. In the path integral formalism the equilibrium properties are calculated by using closed paths and an euclidean coordinate system. We show that the calculations on these paths are coordinates independent. In the change of coordinate systems we consider those preserving the physics on which we focus. Recently it has been shown that the path integral formalism can be built from the real motion of particles. We consider the change of coordinates for which the equations of motion are unchanged. Thus we have to deal with the canonical transformations. The Lagrangian is not uniquely defined and a change of coordinates introduces in hamiltonians the partial time derivative of an arbitrary function. We have show that the closed paths does not contain any arbitrary ingredients. This proof is inspired by a method used in gauge theory. Closed paths appear as the keystone on which we may describe the equilibrium states in statistical thermodynamics.
Highlights
The path integral method has been proposed by Feynman [1] as an alternative to the Schrodinger equation
In (2) we must take xi(0) = xi(βh) showing that we only consider closed paths, this is associated with the fact that in traditional version of statistical mechanics we only focus on the trace of the density matrix
The paths represent the particles trajectories that are quantified via the functional integration in which the Heisenberg uncertainty relations appear. With this approach we do not use the Schrodinger equation, this appears as a necessity if we want to use the same formalism for describing both equilibrium states and time-irreversible processes
Summary
The path integral method has been proposed by Feynman [1] as an alternative to the Schrodinger equation. The path integral formalism or the functional integral point of view has been extended in statistical thermodynamics (see for instance [3]) In quantum physics this formalism has been extensively used but due to the presence of a complex measure it is difficult to have rigorous mathematical treatments. In the initial version of statistical physics the path integral formalism requires to solve the Schodinger equation, to use the canonical form of the density matrix and to introduce some mathematical tricks. Published under licence by IOP Publishing Ltd have to be inspected on a time associated with a non usual equilibrium condition In such an approach the time is a real time and the path integral is not just a formal mathematical trick.
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