Jarzynski Equality for Driven Quantum Field Theories

The fluctuation theorems, and in particular, the Jarzynski equality, are the most important pillars of modern nonequilibrium statistical mechanics. We extend the quantum Jarzynski equality together with the two-time measurement formalism to their ultimate range of validity—to quantum field theories. To this end, we focus on a time-dependent version of scalar ${\varphi }^4$. We find closed-form expressions for the resulting work distribution function, and we find that they are proper physical observables of the quantum field theory. Also, we show explicitly that the Jarzynski equality and Crooks fluctuation theorems hold at one-loop order independent of the renormalization scale. As a numerical case study, we compute the work distributions for an infinitely smooth protocol in the ultrarelativistic regime. In this case, it is found that work done through processes with pair creation is the dominant contribution.

Popular Summary

Since the middle of the last century, quantum field theory—a theoretical framework for describing subatomic particles in terms of fields—has served as our most fundamental description of nature. The theory has been rigorously tested and encompasses particle physics, cosmology, and condensed matter. Thermodynamics, however, remained largely stagnant until about two decades ago when fluctuation theorems broadened our understanding of systems operating far from equilibrium. These two bodies of research remain largely separate. Here, we extend the existing literature on fluctuation theorems to the realm of quantum field theory, vastly expanding the range of possible systems to which these theorems can be applied.

In our work, we reconcile the sometimes conflicting nature of quantum fluctuation theorems and quantum field theory. Quantum fluctuation theorems aim to describe the behavior of systems over short time scales when subject to time-dependent driving, whereas quantum field theories are usually used to describe the long-time behavior of time-independent systems. We require new calculation techniques because existing methods in quantum field theory are inadequate for calculating the quantities of interest. This culminates in the calculation of closed-form expressions for the probability distribution of work performed by subjecting a particular quantum field theory to a time-dependent driving.

From the quark-gluon plasma produced in particle accelerators to the relativistic charge carriers of graphene and to the inflation of the early Universe, nonequilibrium systems described by quantum field theories abound across a vast range of scales. This work opens the door for future applications of quantum fluctuation theorems to the study of this diverse group of systems.

Figure 1

Leading-order corrections to the propagator of the scalar field $\varphi$. The interactions are sourced by insertions of the classical, nondynamical field ${\chi }_{\mathrm{cl}}$.

Figure 2

Vacuum energy contributions of ${\chi }_{\mathrm{cl}}$. The interactions are sourced by insertions of the classical, nondynamical field ${\chi }_{\mathrm{cl}}$.

Figure 3

Diagrams that contribute to the work distribution function ${\rho }_{n\to n+2}(W)$.

Figure 4

Work distribution function and its decomposition into subprocesses for $\beta =m=1$ with driving protocol specified in Eq. (33).

Figure 5

An incomplete collection of possible Feynman diagrams which contribute to the nontrivial part of the work distribution for $n\to n$ scattering. For visual clarity, propagators which are not part of the four-point function are represented with dashed lines and insertions of the background field ${\chi }_{\mathrm{cl}}$ are omitted.

Figure 6

Glued diagrams generated by identifying the field sources of one Feynman diagram in Fig. 5 with the field sources of another and then connecting the diagrams. For visual clarity, propagators which are not part of the four-point function are represented with dashed lines and field sources are denoted by a small square. Insertions of the background field ${\chi }_{\mathrm{cl}}$ are omitted.