Contact geometry and quantum thermodynamics of nanoscale steady states
Graphical abstract
Introduction
Boltzmann and Gibbs formulated the prescription of equilibrium statistical mechanics by providing the appropriate density matrix (or density operator) of a system kept at temperature which is given by the canonical distribution, where (the Boltzmann constant is set to unity throughout the paper) and is the normalizing partition function [1]. For systems in (weak) contact with both a thermostat and a particle reservoir, the grand canonical distribution describes the statistical state of the system. Hence, this canonical or grand canonical prescription is the foundation stone of equilibrium statistical physics and these phase space distributions govern the laws of classical equilibrium thermodynamics. Thus, equilibrium statistical mechanics is all about finding the correct density matrix or the phase space distribution of a system enabling us to compute various equilibrium properties. On the other hand, we can utilize the linear response theory [2], [3] and Onsager reciprocity relationships [4], [5] to study systems close to thermodynamic equilibrium. But, the majority of biological, technological and even cosmological systems fall into the category of systems far from equilibrium where there is no general formalism to obtain correct the density matrix of the system. This active and vibrant research area is primarily centred around transport phenomena [6], [7], chemical transformations [8] and autonomous machines [9], [10] (both classical and quantum perspective). Although thermodynamic relations away from equilibrium in general situations have been explored before (see for example [11], [12], [13], [14], [15]) including situations involving non-interacting transport [16], [17], [18], [19], [20], this still leaves a lot to be formulated, as definitions of basic thermodynamic quantities still remain to be understood. Some important results in truly non-equilibrium steady state situations for certain classes of models have recently been revealed in [21], which will form the basis for our analysis and to which we return in Section 2.
Quite remarkably, it has been found in the recent years that thermodynamic properties are consistent with the quantum properties of nanoscale systems which are far from the thermodynamic limit and may even contain a single particle. The advancement in technologies and experimental techniques have enabled us to measure the thermodynamic properties of microscopic systems [22], [23], [24], [25]. For example Collin et al. [26] measured the work performed on a single RNA hairpin using optical tweezers and also found the equilibrium free energy change using the work fluctuation relations from such out of equilibrium measurements. Moreover, the Jarzynski equality [27], [28], [29], [30] relating different thermodynamic quantities for systems driven far from equilibrium and fluctuation theorems related to entropy production probability during a finite period of time [31], [32], [33], [34] have been verified theoretically as well as experimentally. Interestingly, some of these ideas of non-equilibrium physics find applications in machine learning [35] as well as learning and inference problems [36], [37]. A more profound problem in modern non-equilibrium thermodynamics is the optimization of thermodynamic efficiency of a molecular-scale machine which performs useful work without excessive dissipation, using thermodynamic length [38], [39], [40], [41], [42]. Further extensions of the concept of thermodynamic length for microscopic systems involving the Fisher information matrix can be found in Refs. [43], [44]. Recently, Sivak and Crooks formulated a linear response framework for optimal protocols that minimize dissipation during non-equilibrium perturbations of microscopic systems [45].
A particularly interesting aspect of non-equilibrium thermodynamics ensues while considering systems arbitrarily far from equilibrium (not necessarily in the linear response regime) but working under steady state conditions. Such thermodynamic states are known as non-equilibrium steady states (NESS) [46], [47] and primarily differ from thermodynamic equilibrium states in respect to the fact that there is a non-zero entropy production since they are not equilibrium states. However, the steady nature of the problem implies that entropy is being accumulated at a constant rate. There has been a notably large body of work in the field of NESS recently [46], [48], [49], [50], [51], [52], [53]. The primary aim of this paper is to develop a geometric formulation of non-equilibrium steady states in quantum thermodynamics in parallel to the developments in equilibrium thermodynamics of systems in the thermodynamic limit.
Motivation and plan: Since several decades, contact geometry has been considered to be a suitable framework for the geometric formulation for dissipative mechanical systems and also for thermodynamics [54], [55], [56], [57], [58] with varied motivations [59], [60], [61]. However, most of the developments are confined to the field of equilibrium thermodynamics with thermodynamic transformations corresponding to contact Hamiltonian flows on the Legendre submanifolds [54], [55], [60], [61]. Although there are formalisms such as the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) for dealing with dynamical irreversible situations [62], [63], [64], these need to be developed further for applications. The GENERIC, which describes a general class of dynamics obtained by pasting a symmetric dissipative part to an anti-symmetric Poisson part has been studied from the point of view of contact geometry. It should be pointed out that the GENERIC admits a natural geometric description as a second class metriplectic system [65], [66]. However, such directions of work are different from the point of view adapted in the present paper where the formalism for a class of quantum steady states such as steady state particle transport through a quantum dot closely following the formalism in [21] is developed in parallel to that known for equilibrium thermodynamics. It will be shown that the non-equilibrium thermodynamic phase space has an underlying contact structure associated with it and contact Hamiltonian dynamics can be used to describe transformations between different steady states. We also develop a Hamilton–Jacobi theory where the steady state extension of the Massieu–Planck function takes the role of the principal function. Finally, for systems with exponential reduced density matrices it is shown that the thermodynamic metric (equivalent to the Fisher information matrix) on control parameter spaces as presented in [10], [41] smoothly comes out from our formalism and that other thermodynamic Hessian metrics on control parameter spaces are conformally equivalent to it. This firmly establishes the general nature of the Fisher information matrix and also provides other equivalent ways in which it might be computed.
With this background, the rest of the paper is organized as follows. In the next section, we discuss some basic aspects of steady state quantum thermodynamics of a class of nanoscale systems. We then provide a minimal digression on contact geometry which shall make the paper self contained. Section 3 is devoted to our results on the geometry of quantum NESS. Finally, we shall end with discussions on our results in Section 4.
Section snippets
Basics of NESS & contact geometry
We shall start by briefly describing the notion of steady state quantum thermodynamics of nanoscale systems with a simple example and also set up our notation. This is done in the subsection below. In the subsequent subsection, we describe the basics of contact geometry and Hamiltonian dynamics on contact manifolds.
Geometry of quantum NESS: Some results
We will now show that the non-equilibrium steady states correspond to points on Legendre submanifolds of the non-equilibrium thermodynamic phase space which has the structure of a contact manifold. In this context we propose several results which will further the understanding of the geometric structure of NESS. Recalling the steady state relation immediately leads to the identification that are local coordinates on a contact manifold with . The
Discussion
Investigation of the geometry of the non-equilibrium thermodynamic phase space, based upon the differential geometric approach provides a deeper understanding of the structure of thermodynamics and statistical mechanics. For steady state quantum systems arbitrarily far from equilibrium, we have established that NESS are points on control parameter spaces which are Legendre submanifolds of an ambient non-equilibrium thermodynamic phase space with an underlying contact structure. The structure
CRediT authorship contribution statement
Aritra Ghosh: Analysis, Conceptualization, Preparation of manuscript. Malay Bandyopadhyay: Analysis, Conceptualization, Preparation of manuscript. Chandrasekhar Bhamidipati: Analysis, Conceptualization, Preparation of manuscript.
Declaration of Competing Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Chandrasekhar Bhamidipati reports travel was provided by Science and Engineering Research Board, Government of India. Aritra Ghosh reports financial support was provided by PMRF, Ministry of Human Resource Development, Government of India.
Acknowledgements
We are grateful to Rishabh Raturi for valuable discussions. A.G. would like to acknowledge the support by the International Centre for Theoretical Sciences (ICTS) for the online program - Bangalore School on Statistical Physics - XI (code: ICTS/bssp2020/06) and the financial support received from the M.H.R.D., Government of India in the form of a Prime Minister’s Research Fellowship. M.B. gratefully acknowledges financial support from Department of Science and Technology (DST), India under the
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All authors contributed equally to the manuscript preparation.