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Bakewell-Smith, George (2025) Concentration Inequalities for Fluctuations in Classical and Quantum Markov Processes. PhD thesis, University of Nottingham.

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Abstract

Markov processes describe a plethora of physical systems in nature, including the mechanics of molecular motors, chemical reactions, and even financial markets. In the framework of continuous-time Markov processes, there are many important quantities of interest. One example are dynamical observables: time-integrated functionals of stochastic trajectories. Another are first passage times (FPTs), which are the time taken for an observable to reach a fixed threshold. The fluctuations of these two quantities are bounded from below by the thermodynamic uncertainty relations (TURs), a fundamental result which captures the tradeoff between their precision, and physical parameters such as entropy production or activity. Whilst these lower bounds are well-known, less attention has been given to finding upper bounds on fluctuations.

We first prove the existence of general upper bounds, at any time, on the variance of any linear combination of fluxes for classical, continuous-time Markov processes. These are derived by considering perturbed dynamics and applying techniques in concentration theory, in particular the Cramer-Chernoff method. We call these bounds "inverse thermodynamic uncertainty relations". Spectral methods allow us to express the bounds in terms of parameters of the dynamics which include the symmetrised spectral gap of the generator and max/min escape rates, alongside observable-dependent quantities. Afterwards we provide a concentration inequality for dynamical observables, which upper bounds the probability distribution for finite time. We then extend these results to FPTs for the dynamical activity, as well as a tail bound for general counting observables. Finally we generalise our FPT results to the quantum framework for general quantum Markov processes and reset processes, including a tail bound for FPTs of general counts, i.e. counts of a subset of emissions. We also prove a large deviation principle for FPTs of classical and quantum counting processes.

Our results have several consequences and applications. By providing upper bounds on the relative uncertainty, the range of estimation errors is bounded from both sides. These findings suggest that spectral quantities limit the range of fluctuations. The observable-dependent parameters in the bounds offer an advantage over the traditional TURs for precision estimation. Inverse TURs provide bounds on the accuracy of classical or quantum clocks or the efficiency of heat engines. The concentration inequalities can be used in finite-regime parameter estimation of classical and quantum processes, with open quantum dynamics being particularly relevant in experimental situations.

Item Type:	Thesis (University of Nottingham only) (PhD)
Supervisors:	Guta, Madalin Garrahan, Juan P.
Keywords:	Classical Markov chains, Quantum Markov processes, Counting observables, First passage times, Large deviations, Concentration bounds, Thermodynamic uncertainty relations, Markov processes
Subjects:	Q Science > QA Mathematics > QA273 Probabilities
Faculties/Schools:	UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID:	81013
Depositing User:	Bakewell-Smith, George
Date Deposited:	31 Jul 2025 04:40
Last Modified:	31 Jul 2025 04:40
URI:	https://eprints.nottingham.ac.uk/id/eprint/81013

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