Thermodynamics and Energy

The Unlikelihood of Stochastic Paths in Multidimensional Phase Spaces

Authors: Arturo Tozzi
Comments: 8 Pages.

The methodological/operational virtues of stochastic approaches allow us to assess biophysical phenomena in terms of random walks, Brownian motions, Markov chains, etc. We argue that these approaches are not profitable when stochastic paths occur in high-dimensional phase spaces. Indeed, contrary to the two-dimensional random walks, the higher-dimensional random walks do not resume to the starting point: the more the phase space’s dimension, the more the (seemingly) stochastic paths are confined/constrained. The mathematical impossibility of high-dimensional random paths has numerous implications, both epistemological and operational. Stochasticity should no longer be used for the evaluation of real-world systems, since the experimental assessment of multifactorial biophysical phenomena requires numerous parameters, each one standing for a different dimension in the phase space. Furthermore, multidimensional trajectories cannot generate circular, recurrent paths returning to the starting point, making unnecessary powerful methodological weapons correlated with cyclic configurations such as the Jordan curve theorem and the Betti number. Since higher-dimensional trajectories are unable to cross all the microstates with the same probability, our account suggests the unlikelihood of ergodic paths in multidimensional phase spaces, casting also doubts on the Shannon’s account of information entropy in the continuous case. Next, we describe how the memory of old events is preserved in high-dimensional phase spaces, since memoryless events disconnected from the past, e.g., stochastic resetting and Markov chains, are banned. We want to end up with an epistemological consideration. The Heisenberg’s uncertainty principle does not allow to measure at the same time both the position and the velocity of a quantum object. The chaotic logistic map does not allow to know exactly the particle location when the phase parameter is between 3 and 4. The Godel’s theorem does not allow to find statements about natural numbers that are at the same time true and provable within a consistent formal system. In these three examples from quantum dynamics, nonlinear dynamics and mathematical foundations, the unfeasibility to reach a single univocal result and to express simultaneously well-defined conjugate properties by a single value does depend on neither technical difficulties, nor failures in the observational devices of the current technology, nor our ignorance of some fundamental property of reality, rather it stands for an insurmountable, irreducible, universal feature that is intrinsic to the system under evaluation and cannot be even in theory solved. In our case, too, the unfeasibility of random paths in high dimensions stands for an intrinsic mathematical feature of the system, an epistemological boundary that cannot be overtaken by methodological tricks. Once expunged the randomness from the evaluation of multiparametric phenomena, the sole approach to partially rescue its methodological/operational virtues is to limit ourselves to the study of oversimplified, lower-dimensional systems.
Category: Thermodynamics and Energy