Uncertainty Relations in Stochastic Systems

Singh, Dipesh Kumar (2025) Uncertainty Relations in Stochastic Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Dipesh Kumar Singh (20MS176)) 20MS176_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (7MB)

Abstract

In this thesis, we design a thermal bath that preserves the conservation of a system’s angular momentum or allows it to fluctuate around a specified nonzero mean while maintaining a Boltzmann distribution of energy in the steady state. We demonstrate that classical particles immersed in such baths exhibit position-momenta uncertainties with a strictly positive lower bound proportional to the absolute value of the mean angular momentum. The proportionality constant, c, is dimensionless and attains the exact value c = 1 2 for particles in central potentials. We consider limit cycle systems as an example where we can realize these uncertainty bounds. We show that limit cycle systems in Langevin bath exhibit uncertainty in observables that define the limit-cycle plane, and maintain a positive lower bound. The uncertainty-bound depends on the parameters that determine the shape and periodicity of the limit cycle. In one dimension, we use the framework of canonical dissipative systems to construct the limit cycle, whereas in two dimensions, the conserved angular momentum ensemble itself is a limit cycle system under temperature. We also investigate how uncertainties, which are absent in deterministic systems, increase with time when the systems are attached to a bath and eventually cross the lower bound before reaching the steady state. We further lay out the foundational work for studying non-central potentials, higher dimensional as well as multiparticle interacting systems in the constructed ensembles. v

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