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We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results.

Nonequilibrium work relations and fluctuation theorems permit us to extract equilibrium information from nonequilibrium measurements. They find application in single-molecule pulling experiments where molecular free energies can be determined from irreversible work measurements by using unidirectional (e.g., Jarzynski’s equality) and bidirectional (e.g., Crooks fluctuation theorem and Bennet’s acceptance ratio (BAR)) methods. However, irreversibility and the finite number of pulls limit their applicability: the higher the dissipation, the larger the number of pulls necessary to estimate ΔG within a few kBT. Here, we revisit pulling experiments on an RNA three-way junction (3WJ) that exhibits significant dissipation and work-distribution long tails upon mechanical unfolding. While bidirectional methods are more predictive, unidirectional methods are strongly biased. We also consider a cyclic protocol that combines the forward and reverse work values to increase the statistics of the measurements. For a fixed total experimental time, faster pulling rates permit us to efficiently sample rare events and reduce the bias, compensating for the increased dissipation. This analysis provides a more stringent test of the fluctuation theorem in the large irreversibility regime.

Ted Jacobson discovered that gravity was related to thermodynamics. However, the calculated temperature using the Boltzmann area entropy is still not reasonable. We searched and discovered an empirical equation for the gravitational constant with a reasonable temperature. The calculated value was 3.20 K, which is similar to the temperature of the cosmic microwave background of 2.73 K. Then, we examined Yasuo Katayama’s theory. For this purpose, we introduced the modified Wagner’s equation, which is compatible with Jarzynski equality. Finally, using Ted Jacobson’s theory, we proposed our theory of gravity with the Gibbs volume entropy.

We report equilibrium and nonequilibrium molecular dynamics (MD) simulations of two nucleosome core particles (NCPs) stacked with their dyad axes oriented in parallel or antiparallel fashion. From the equilibrium trajectories, we determine the bridging behavior of different histone tails and observe that different sets of histone tails play important roles in the two orientations in stabilizing the NCP stack. While the H4 and H2A tails play important intermediary roles in the parallel stack, the H3 and H2B tails are important in the antiparallel stack. We use steered MD simulations to unstack the two NCPs and find a stark difference in their unstacking pathways. While the average rupture force was found to be higher for the parallel stack, the work done for complete unstacking was similar for both orientations. We use Jarzynski equality to determine the PMF profiles along the unstacking pathway, relate our findings to the behavior of NCP mesophases, and derive insights into the enigmatic nucleosomal organization in the chromatin fiber.

Work is an important quantity in thermodynamics. In a closed quanutm system, the two-point energy measurements can be applied to measure the work but cannot be utilized in an open quantum system. With the two-point energy measurements, it has been shown that the work fluctuation satisfies the Jarzynski equality. We propose a scheme to measure the work in an open quantum system through the technique of reservoir engineering. Based on this scheme, we show that the work fluctuation in open quantum system may violate the Jarzynski equality. We apply our scheme to a two-level atom coupled to an engineered reservoir and numerically justify the general results, especially demonstrating that the second law of thermodynamics can be violated.

In this paper, we will obtain quantum work for a quantum scale five dimensional Myers-Perry black hole. Unlike heat represented by Hawking radiation, the quantum work is represented by a unitary information preserving process, and becomes important for black holes only at small quantum scales. It will be observed that at such short distances, the quantum work will be corrected by non-perturbative quantum gravitational corrections. We will use the Jarzynski equality to obtain this quantum work modified by non-perturbative quantum gravitational corrections. These non-perturbative corrections will also modify the stability of a quantum Myers-Perry black hole. We will define a quantum corrected information geometry by incorporating the non-perturbative quantum corrections in the information geometry of a Myers-Perry black hole. We will use several different quantum corrected effective information metrics to analyze the stability of a quantum Myers-Perry black hole., 22, pages, 9 figures. Accepted for publication in JHEP

The generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics was theoretically investigated in the context of non-equilibrium statistical mechanics. The recent study of the authors revealed that the Loewner evolution enables encoding the non-equilibrium (irreversible) processes into equilibrium (reversible) processes. In this study, by Gibbs entropy-based information-theoretic approaches, we formulated this encoding mechanism of the SLE to discuss its advantages as a mean to better describe non-equilibrium states. After deriving entropy production and flux for the 2D trajectories of the generalized SLE curve, we reformulated the system's entropic properties in terms of the Kullback-Leibler (KL) divergence. We demonstrate that this operation leads to alternative expressions of the Jarzynski equality and the second law of thermodynamics, which are consistent with the previously suggested theory of information thermodynamics. The irreversibility of the 2D trajectory was likewise discussed by decomposing its entropy into additive and non-additive parts. We numerically verified the non-equilibrium property of our model by simulating the long-time behavior of the entropic measure suggested by our formulation, referred to as the relative Loewner entropy., 25 pages, 2 figures

We show that the most prominent of the work theorems, the Jarzynski equality and the Crooks relation, can be applied to the momentum transfer at the air–sea interface using a hierarchy of local models. In the more idealized models, with and without a Coriolis force, the variability is provided from Gaussian white noise which modifies the shear between the atmosphere and the ocean. The dynamics is Gaussian, and the Jarzynski equality and Crooks relation can be obtained analytically solving stochastic differential equations. The more involved model consists of interacting atmospheric and oceanic boundary layers, where only the dependence on the vertical direction is resolved, the turbulence is modeled through standard turbulent models and the stochasticity comes from a randomized drag coefficient. It is integrated numerically and can give rise to a non-Gaussian dynamics. Also in this case the Jarzynski equality allows for calculating a dynamic beta βD of the turbulent fluctuations (the equivalent of the thermodynamic beta β=(kBT)-1 in thermal fluctuations). The Crooks relation gives the βD as a function of the magnitude of the work fluctuations. It is well defined (constant) in the Gaussian models and can show a slight variation in the more involved models. This demonstrates that recent concepts of stochastic thermodynamics used to study micro-systems subject to thermal fluctuations can further the understanding of geophysical fluid dynamics with turbulent fluctuations.

The free energy of glasses cannot be estimated using thermodynamic integration, as glasses are intrinsically not in equilibrium. We present numerical simulations showing that, in contrast, plausible free-energy estimates of a Kob-Andersen glass can be obtained using the Jarzynski relation. Using the Jarzynski relation, we also compute the chemical potential difference of the two components of this system, and find that, in the glassy regime, the Jarzynski estimate matches well with the extrapolated value of the supercooled liquid. Our findings are of broader interest as they show that the Jarzynski method can be used under conditions where the thermodynamic integration approach, which is normally more accurate, breaks down completely. Systems where such an approach might be useful are gels and jammed, glassy structures formed by compression., 4 pages, 2 figures

We study the work distribution of a Brownian particle diffusing in an environment of active particles and being trapped in a harmonic potential, the center of which is subjected to a time-dependent protocol. Employing phase space path integral technique we find an expression of work distribution for any generic model of active noise. Here we consider two active noise models — Gaussian correlated and Poisson white, each of which can represent some physical systems. For both the cases, it is found that transient fluctuation relation of work is not applicable though at steady state it holds by defining a renormalized temperature τ r in place of bath temperature. Interestingly, τ r is the same for both the models and can be expressed in terms of diffusivities of active and thermal noises. For correlated Gaussian bath, an alternative approach is presented. Analogous to the formalism given by Hatano and Sasa (2001), we obtain a work like quantity from nonequilibrium potential with the inclusion of a new stationary parameter Ω . With proper choice of Ω , a steady-state fluctuation relation, namely Jarzynski equality is satisfied.

The Jarzynski equality is one of the most widely celebrated and scrutinized nonequilibrium work theorems, relating free energy to the external work performed in nonequilibrium transitions. In practice, the required ensemble average of the Boltzmann weights of infinite nonequilibrium transitions is estimated as a finite sample average, resulting in the so-called Jarzynski estimator, (Formula presented.). Alternatively, the second-order approximation of the Jarzynski equality, though seldom invoked, is exact for Gaussian distributions and gives rise to the Fluctuation-Dissipation estimator (Formula presented.). Here we derive the parametric maximum-likelihood estimator (MLE) of the free energy (Formula presented.) considering unidirectional work distributions belonging to Gaussian or Gamma families, and compare this estimator to (Formula presented.). We further consider bidirectional work distributions belonging to the same families, and compare the corresponding bidirectional (Formula presented.) to the Bennett acceptance ratio ((Formula presented.)) estimator. We show that, for Gaussian unidirectional work distributions, (Formula presented.) is in fact the parametric MLE of the free energy, and as such, the most efficient estimator for this statistical family. We observe that (Formula presented.) and (Formula presented.) perform better than (Formula presented.) and (Formula presented.), for unidirectional and bidirectional distributions, respectively. These results illustrate that the characterization of the underlying work distribution permits an optimal use of the Jarzynski equality. Fil: Arrar, Mehrnoosh. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Química, Física de los Materiales, Medioambiente y Energía. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Química, Física de los Materiales, Medioambiente y Energía; Argentina Fil: Boubeta, Fernando Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Química, Física de los Materiales, Medioambiente y Energía. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Química, Física de los Materiales, Medioambiente y Energía; Argentina Fil: Szretter Noste, María Eugenia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina Fil: Sued, Raquel Mariela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Boechi, Leonardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina Fil: Rodriguez, Daniela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina

In this work we explore the validity of the transient work fluctuation theorem as well as the Jarzynski equality. In the case of a Brownian particle dragged through a fluid by an optical trap, the fluid plays the role of a non-Markovian heat bath characterized by an specific Ornstein–Uhlenbeck friction memory kernel. To achieve our goal, we used an alternative method which transformed the generalized Langevin equation into an equivalent third-order non-Markovian Langevin equation. By means of the exact analytical solution of this Langevin equation, we calculated the statistics of thermodynamic work which established the Crooks relation, and then the Jarzynski equality came immediately. The fluctuation theorems are applicable to small systems where energy fluctuations with respect to the average behavior are big enough to be measured. The examples of such a class of systems are found inside living organisms.

A system--bath (SB) model is considered to examine the Jarzynski equality in the fully quantum regime. In our previous paper [J. Chem. Phys. 153 (2020) 234107], we carried out "exact" numerical experiments using hierarchical equations of motion (HEOM) in which we demonstrated that the SB model describes behavior that is consistent with the first and second laws of thermodynamics and that the dynamics of the total system are time irreversible. The distinctive quantity in the Jarzynski equality is the "work characteristic function (WCF)", $\langle \exp[-\beta W] \rangle$, where $W$ is the work performed on the system and $\beta$ is the inverse temperature. In the present investigation, we consider the definitions based on the partition function (PF) and on the path, and numerically evaluate the WCF using the HEOM to determine a method for extending the Jarzynski equality to the fully quantum regime. We show that using the PF-based definition of the WCF, we obtain a result that is entirely inconsistent with the Jarzynski equality, while if we use the path-based definition, we obtain a result that approximates the Jarzynski equality, but may not be consistent with it., Comment: Open access

The Jarzynski relation, as an equality form of the second law of thermodynamics, represents an exact thermodynamic statement that is valid arbitrarily far away from equilibrium. This remarkable relation directly links the equilibrium free energy difference between two states and the probability distribution of the work done along a process that drives the system from one state to the other. Here, we leverage the Jarzynski equality and a local equilibrium assumption, to go beyond the calculation of free energy differences and also extract the dissipation potential from additional measurements of kinematic field variables (strain and velocity fields). The proposed strategy is exemplified over pulling experiments of mass–spring models obeying overdamped Langevin dynamics, which is a prototype for nanorods, fibrous macro-molecules and the Rouse model of polymers. Different interaction potentials, fluid viscosities and bath temperatures are studied, so as to intrinsically control how close or far away the system is from equilibrium. The data-inferred continuum models are then validated against processes governed by different pulling protocols, thereby demonstrating their predictive capability. The methods presented here represent a first step toward full material characterization from non-equilibrium data of macroscopic observables, which could potentially be obtained from experimental observations.

We explore the role a non-Markovian memory kernel plays on information exchange and entropy production in the context of a external work protocol. The Jarzynski Equality is shown to hold for both the harmonic and the non-harmonic models. We observe the memory function acts as an information pump, recovering part of the information lost to the thermal reservoir as a consequence of the non-equilibrium work protocol. The pumping action occurs for both the harmonic and non-harmonic cases. Unexpectedly, we found that the harmonic model does not produce entropy, regardless of the work protocol. The presence of even a small amount of non-linearity recovers the more normal entropy producing behaviour, for out-of-equilibrium protocols., 32 pages, 18 figures

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks’ fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.

We introduce a simple enhanced sampling approach for the calculation of free energy differences and barriers along a one-dimensional reaction coordinate. First, a small number of short nonequilibrium simulations are carried out along the reaction coordinate, and the Jarzynski equality is used to learn an approximate free energy surface from the nonequilibrium work distribution. This free energy estimate is represented in a compact form as an artificial neural network and used as an external bias potential to accelerate rare events in a subsequent molecular dynamics simulation. The final free energy estimate is then obtained by reweighting the equilibrium probability distribution of the reaction coordinate sampled under the influence of the external bias. We apply our reweighted Jarzynski sampling recipe to four processes of varying scales and complexities.spanning chemical reaction in the gas phase, pair association in solution, and droplet nucleation in supersaturated vapor. In all cases, we find reweighted Jarzynski sampling to be a very efficient strategy, resulting in rapid convergence of the free energy to high precision.

Agradecimentos: This work is supported by the U.S. Department of Energy, the Laboratory Directed Research and Development (LDRD) program and the Center for Nonlinear Studies at LANL. AS offers his gratitude to Yi-Xiang Liu and Paola Cappellaro for insightful discussions Abstract: It has been established that the inclusive work for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the conditional stochastic work for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the energy conditioned on the initial energy surface. As main results, we obtain a generalized Jarzynski equality and a sharper maximum work theorem, which account for how non-adiabatic the process is. Our findings are illustrated with the parametric harmonic oscillator Fechado

The quantum version of Jarzynski equality was theoretically investigated for the work operator in the interaction picture. Since the Hamiltonian is time-dependent, it does not commute with itself at different times, then we have solved analytically its dynamics with lie-algebraic techniques. The physical object is a superconducting optical cavity. It was shown that the quantum dynamics of a cavity does not obey Jarzynski’s equality in the small temperature regime $$(\beta \omega _0 \hbar \gg 1)$$ , if we consider a quantum work operator defined in a protocol interaction picture. The case of higher temperatures $$(\beta \omega _0 \hbar \ll 1)$$ we are in Jarzynski regime and Jarzynski equality can be used to obtain statistical information about work done on the cavity. Since, for all used parameters, the state is a thermal state and its dynamics is of the harmonic oscillator, then this quantum regime is related to the protocol action. The protocol can be implemented by injecting a field in the cavity, thus it creates quantum resources.

Thermodynamics with multiple conserved quantities offers a promising direction for designing novel devices. For example, Vaccaro and Barnett's [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011)1364-502110.1098/rspa.2010.0577; S. M. Barnett and J. A. Vaccaro, Entropy 15, 4956 (2013)ENTRFG1099-430010.3390/e15114956] proposed information erasure scheme, where the cost of erasure is solely in terms of a conserved quantity other than energy, allows for new kinds of heat engines. In recent work, we studied the discrete fluctuations and average bounds of the erasure cost in spin angular momentum. Here we clarify the costs in terms of the spin equivalent of work, called spinlabor, and the spin equivalent of heat, called spintherm. We show that the previously found bound on the erasure cost of γ^{-1}ln2 can be violated by the spinlabor cost, and only applies to the spintherm cost. We obtain three bounds for spinlabor for different erasure protocols and determine the one that provides the tightest bound. For completeness, we derive a generalized Jarzynski equality and probability of violation which shows that for particular protocols the probability of violation can be surprisingly large. We also derive an integral fluctuation theorem and use it to analyze the cost of information erasure using a spin reservoir.

A fundamental problem in thermodynamics is the recovery of macroscopic equilibrated interaction energies from experimentally measured single-molecular interactions. The Jarzynski equality forms a theoretical basis in recovering the free energy difference between two states from exponentially averaged work performed to switch the states. In practice, the exponentially averaged work value is estimated as the mean of finite samples. Numerical simulations have shown that samples having thousands of measurements are not large enough for the mean to converge when the fluctuation of external work is above 4 kBT, which is easily observable in biomolecular interactions. We report the first example of a statistical gamma work distribution applied to single molecule pulling experiments. The Gibbs free energy of surface adsorption can be accurately evaluated even for a small sample size. The values obtained are comparable to those derived from multi-parametric surface plasmon resonance measurements and molecular dynamics simulations., Measuring interaction energies from experimentally measured single-molecular interactions is challenging. Here, the authors report a gamma work distribution applied to single molecule pulling events for estimating peptide absorption free energy.