{'fraction_non_alphanumeric': 0.027777777777777776, 'fraction_numerical': 0.0, 'mean_word_length': 5.0590909090909095, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Non-equilibrium Annealed Damage Phenomena: A Path Integral Approach

We investigate the applicability of the path integral of non-equilibrium statistical mechanics to non-equilibrium damage phenomena. As an example, a fiber-bundle model with a thermal noise and a fiber-bundle model with a decay of fibers are considered. Initially, we develop an analogy with the Gibbs formalism of non-equilibrium states. Later, we switch from the approach of non-equilibrium states to the approach of non-equilibrium paths. Behavior of path fluctuations in the system is described in terms of effective temperature parameters. An equation of path as an analogue of the equation of state and a law of path-balance as an analogue of the law of conservation of energy are developed. Also, a formalism of a free energy potential is developed. For fluctuations of paths in the system, the statistical distribution is found to be Gaussian. Also, we find the ‘true’ order parameters linearizing the matrix of fluctuations. The last question we discuss is the applicability of the phase transition theory to non-equilibrium processes. From near-equilibrium processes to stationary processes (dissipative structures), and then to significantly non-equilibrium processes: Through these steps we generalize the concept of a non-equilibrium phase transition.