Classical field theory and stochastic properties of hyperbolic equations of dissipative processes

The set of damped hyperbolic transport equations is one of the wide class of equations for the description of dissipative physical processes. Deeper understanding into the structure of these physical phenomena can be obtained with the help of the Hamiltonian formalism. In the present paper, we show that the Hamilton–Lagrange formalism can be constructed for these kinds of transport equations. We obtain the Hamiltonian, the canonically conjugate quantities and the Poisson-bracket expressions for them. With this formalism we analyze the statistical properties of path fluctuations in the new conjugated thermodynamic variable space. We show that for short times the stochastic behavior under this new scope obeys the Chapman–Kolmogorov relationship.