We consider the quantum mechanical generalization of Crooks Fluctuation Theorem and Jarzynski Equality for an open quantum system. The explicit expression for microscopic work for an arbitrary prescribed protocol is obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum Jarzynski Equality and their classical counterparts are clarified. Numerical simulations based on a two-level toy model are used to demonstrate the validity of the quantum version of the two theorems beyond linear response theory regime.
For classical systems driven out of equilibrium, Crooks derived a relation (the Crooks-Jarzynski relation), whose special cases include a relation (the Crooks relation) equivalent to the Kawasaki non-linear response relation. We derive a quantum extension of the Crooks-Jarzynski relation without explicitly using the principle of microreversibility. Its special cases lead to the Jarzynski equality and the standard linear response theory with a Green-Kubo formula with a canonical correlation function. We also derive a quantum extension of the Crooks relation using the principle of microreversibi…
By taking full advantage of the dynamic property imposed by the detailed balance condition, we derive a new refined unified fluctuation theorem (FT) for general stochastic thermodynamic systems. This FT involves the joint probability distribution functions of the final phase space point and a thermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and the FTs of heat as well as the trajectory entropy production can be regarded as special cases of this refined unified FT, and all of them are generalized to arbitrary initial distributions. We also find that the refined unified FT…
It has recently been argued that a self-consistency condition involving the Jarzynski equality (JE) and the Crooks fluctuation theorem (CFT) is violated for a simple Brownian process [L. Y. Chen, J. Chem. Phys. 129, 091101 (2008)]. This note adopts the definitions in the original formulation of the JE and CFT and demonstrates the contrary.
There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, relationship between these theorems was investigated by Crooks for the classical stochastic systems. In this letter, we derive quantum analogues of fluctuation theorem and Jarzynski equality microscopic reversibility condition. In other words, …
We obtain the Crooks and the Jarzynski non-equilibrium fluctuation relations using a direct quantum-mechanical approach for a finite system that is either isolated or coupled not too strongly to a heat bath. These results were hitherto derived mostly in the classical limit. The two main ingredients in the picture are the time-reversal symmetry and the application of the first law to the case where an agent performs work on the system. No further assumptions regarding stochastic or Markovian behavior are necessary, neither a master equation or a classical phase-space picture are required. The s…
The estimate of free energy changes based on Bennett's acceptance ratio method is examined in several limiting cases and compared with other estimates based on the Jarzynski equality and on the Crooks relation. While the absolute amount of dissipated work, defined as the surplus of average work over the free energy difference, limits the practical applicability of Jarzynski's and Crooks' methods, the reliability of Bennett's approach is restricted by the difference of the dissipated works in the forward and the backward process. We illustrate these points by considering a Gaussian chain and a …
The target paper presents an experimental verification of a "Jarzynski-related" equality. We show that the latter equality is in fact not related to the Jarzynski equality.
We prove the Jarzynski relation for general stochastic processes including non-Markovian systems with memory. The only requirement for our proof is the existence of a stationary state, therefore excluding non-ergodic systems. We then show how the concepts of stochastic thermodynamics can be used to prove further exact non-equilibrium relations like the Crooks relation and the fluctuation theorem on entropy production for non-Markovian dynamics.
We present a derivation of the Jarzynski identity and the Crooks fluctuation theorem for systems governed by deterministic dynamics that conserves the canonical distribution such as Hamiltonian dynamics, Nose-Hoover dynamics, Nose-Hoover chains and Gaussian isokinetic dynamics. The proof is based on a relation between the heat absorbed by the system during the non-equilibrium process and the Jacobian of the phase flow generated by the dynamics.
In this paper, we address an important question of the relationship between fluctuation theorems for the dissipated work $W_{d} = W-ΔF$ with general finite-time (like Jarzynski equality and Crooks relation) and infinite-time (like Gallavotti-Cohen theorem) drive protocols and their time-reversal symmetric versions. The relations between these kinds of fluctuation relations are uncovered based on the examples of a classical Markovian $N$-level system. Further consequences of these relations are discussed with respect to the possible experimental verifications.
The Jarzynski Equality relates the free energy difference between two equilibrium states of a system to the average of the work over all irreversible paths to go from one state to the other. We claim that the derivation of this equality is flawed, introducing an ad hoc and unjustified weighting factor which handles improperly the heat exchange with a heat bath. Therefore Liphardt et al's experiment cannot be viewed as a confirmation of this equality, although the numerical deviations between the two are small. However, the Jarzynski Equality may well be a useful approximation, e.g. in measurem…
Jarzynski equality [Phys. Rev. E {\bf 56}, 5018 (1997)] is found to be valid with slight modefication for the transitions between nonequilibrium stationary states, as well as the one between equilibrium states. Also numerical results confirm its validity. Its relevance for nonequilibrium thermodynamics of the operational formalism is discussed.
The connection between work and changes in the Hamiltonian for a system with a time-dependent Hamiltonian has recently been called into question, casting doubt on the usefulness of the Jarzynski equality for calculating free energy changes. In this paper, we discuss the relationship between two possible definitions of free energy and show how some recent disagreements regarding the applicability of the Jarzynski equality are the result of different authors using different definitions of free energy. Finally, in light of the recently raised doubts, we explicitly demonstrate that it is indeed po…
It has recently been shown that the Jarzynski equality gets modified, when there are experimental errors in computing work. This modified result also holds good in presence of feedback. In this work, we use a simple toy model, that of a Szilard engine, to prove these results both in the presence and in absence of feedback.
A simple model featuring a double well potential is used to represent a liquid that is quenched from an ergodic state into a history dependent glassy state. Issues surrounding the application of the Jarzynski Equality to glass formation are investigated. We demonstrate that the Jarzynski Equality gives the free energy difference between the initial state and the state we would obtain if the glass relaxed to true thermodynamic equilibrium. We derive new variations of the Jarzynski Equality which are relevant to the history dependent glassy state rather than the underlying equilibrium state. It …
We apply path integrals to study nonequilibrium work theorems in the context of Brownian dynamics, deriving in particular the equations of motion governing the most typical and most dominant trajectories. For the analytically soluble cases of a moving harmonic potential and a harmonic oscillator with time-dependent natural frequency, we find such trajectories, evaluate the work-weighted propagators, and validate Jarzynski's equality.
We have experimentally checked the Jarzynski equality and the Crooks relation on the thermal fluctuations of a macroscopic mechanical oscillator in contact with a heat reservoir. We found that, independently of the time scale and amplitude of the driving force, both relations are satisfied. These results give credit, at least in the case of Gaussian fluctuations, to the use of these relations in biological and chemical systems to estimate the free energy difference between two equilibrium states. An alternative method to estimate of the free nergy difference in isothermal process is proposed t…
Crook's Fluctuation Theorem and Jarzynski equality are immensely powerful tools in obtaining equilibrium properties through non-equilibrium transition between two equilibrium states. In this letter, we propose an extension to the Crook's fluctuation theorem for transition between two non-equilibrium steady states (NESS). Using the proposed theorem, we show that it is possible to obtain free energy differences of multiple equilibrium states from a single set of data obtained from the transition between two NESS. The results are verified using numerical simulations by employing Nose-Hoover dynam…
Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski's equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo's work ({\em Proc. Natl. Acad. Sci. USA} {\bf 98}(7), 3658--3661 (2001)) and Hatano-Sasa equality of steady state thermodynamics ({\em Phys. Rev. Lett.} {\bf 86}, 3463--3466 (2001)) to the general multidimensional case.
We give a perturbative analysis of Crooks relation and Jarzynski equality in an arbitrary driven quantum system weakly coupled to a heat bath. Invoking no efficient Hamiltonian nor any restriction on the form of the coupling, we derive the first-order correction to Crooks relation and Jarzynski equality in terms of the interacting Hamiltonian. A Crooks type of relation about energy increment in the system and nonvanishing heat expenditure to the heat bath up to the second order of interaction would be given by the way. Our results tell us that deviation of the quantum fluctuation relations is …
Simple example: During the sudden expansion of an isolated ideal gas from a small volume V_0 into a larger one V_1, the entropy changes by Delta S= N ln(V_1/V_0)=-beta Delta F>0 but no work W is produced nor absorbed. Consequently, Jarzynski's identity <exp(-beta W)>= exp(-beta Delta F) is wrong.
In this short communication, I give a very simple derivation of the Jarzynski equality, which allows to compute the free energy difference of a body, which is driven between two equilibrium states $A$ and $B$ by an external (time-dependent) force, from the probability distribution function of the work done on the system, regardless of the nature of the transformation (reversible or irreversible) between the states $A$ and $B$. It perhaps throws some light on the debate recently started by Cohen and coworkers, and perhaps explains why all the experiments which has been performed in order to tes…
A quantum analogue of the Jarzynski equality is constructed. This equality connects an ensemble average of exponentiated work with the Helmholtz free-energy difference in a nonequilibrium switching process subject to a thermal heat bath. To confirm its validity in a practical situation, we also investigate an open quantum system that is a spin 1/2 system with a scanning magnetic field interacting with a thermal heat bath. As a result, we find that the quantum analogue functions well.
In the global framework of finding an axiomatic derivation of nonequilibrium Statistical Mechanics from fundamental principles, such as the maximum path entropy -- also known as Maximum Caliber principle -- , this work proposes an alternative derivation of the well-known Jarzynski equality, a nonequilibrium identity of great importance today due to its applications to irreversible processes: biological systems (protein folding), mechanical systems, among others. This equality relates the free energy differences between two equilibrium thermodynamic states with the work performed when going bet…
The crucial condition in the derivation of the Jarzynski equality (JE) from the fluctuation theorem is that the time integral of the phase space contraction factor can be exactly expressed as the entropy production resulting from the heat absorbed by the system from the thermal bath. For the system violating this condition, a more general form of JE may exist. This existence is verified by three \textit{Gedanken} experiments and numerical simulations, and may be confirmed by the real experiment in the nanoscale.
The Jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control. The new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control. Our results lead to a generalized fluctuation-dissipation theorem that reflects the readout information, and can be experimentally tested using small thermodynamic systems. We illustrate our general results by an introducing "information ratchet," which can transport a Brownian particle in one direction and extract …
A single bit memory system is made with a brownian particle held by an optical tweezer in a double-well potential and the work necessary to erase the memory is measured. We show that the minimum of this work is close to the Landauer's bound only for very slow erasure procedure. Instead a detailed Jarzynski equality allows us to retrieve the Landauer's bound independently on the speed of this erasure procedure. For the two separated subprocesses, i.e. the transition from state 1 to state 0 and the transition from state 0 to state 0, the Jarzynski equality does not hold but the generalized versi…
The Jarzynski equality (JE) is known as an exact identity for nonequillibrium systems. The JE was originally formulated for isolated and isothermal systems, while Adib reported an JE extended to an isoenergetic process. In this paper, we extend the JE to an energy-controlled system. We make it possible to control the instantaneous value of the energy arbitrarily in a nonequilibrium process. Under our extension, the new JE is more practical and useful to calculate the number of states and the entropy than the isoenergetic one. We also show application of our JE to a kind of optimization problem…
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 Jarz…
The Jarzynski Equality is a well-known and widely used identity, relating the free energy difference between two states of a system to the work done over some arbitrary, nonequilibrium transformation between the two states. Despite being valid for both stochastic and deterministic systems, we show that the optimal transformation protocol for the deterministic case seems to differ from that predicated from an analysis of the stochastic dynamics. In addition, it is shown that for certain situations, more dissipative processes can sometimes lead to better numerical results for the free energy dif…
The probability densities of work that can be exerted on a quantum system initially staying in thermal equilibrium are constrained by the fluctuation relations of Jarzynski and Crooks, when the work is determined by two projective energy measurements. We investigate the question whether these fluctuation relations may still hold if one employs generalized energy measurements rather than projective ones. Restricting ourselves to a class of universal measurements which are independent of several details of the system on which the work is done, we find sets of necessary and sufficient conditions …
One particle in a classical perfect gas is driven out of equilibrium by changing its mass over a short time interval. The work done on the driven particle depends on its collisions with the other particles in the gas. This model thus provides an example of a non-equilibrium process in a system (the driven particle) coupled to an environment (the rest of the gas). We calculate the work done on the driven particle and compare the results to Jarzynski's equality relating a non-equilibrium work process to an equilibrium free-energy difference. The results for this model are generalised to the case…
The Jarzynski equality relates the free energy difference between two equilibrium states to the fluctuating irreversible work afforded to switch between them. The prescribed fixed temperature for the equilibrium states implicitly constrains the dissipative switching process that can take the system far from equilibrium. Here, we demonstrate theoretically and experimentally that such a relation also holds for the nonisothermal case, where the initial stationary state is not in equilibrium and the switching is effected by dynamically changing temperature gradients instead of a conservative force…
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 …
In this paper, we derive the Jarzynski equality (JE) for an isolated quantum system in three different cases: (i) the full evolution is unitary with no intermediate measurements, (ii) with intermediate measurements of arbitrary observables being performed, and (iii) with intermediate measurements whose outcomes are used to modify the external protocol (feedback). We assume that the measurements will involve errors that are purely classical in nature. Our treatment is based on path probability in state space for each realization. This is in contrast to the formal approach based on projection op…
The Jarzynski equality (JE) provides a nonequilibrium method to measure and calculate the free energy difference (FED). Note that if two systems share the same Hamiltonian at two equilibrium states, respectively, they share the same FED between these two equilibrium states as well. Therefore the calculation of the FED of a system may be facilitated by considering instead another virtual system designed to this end. Taking advantage of this flexibility and the JE, we show that by introducing an integrable virtual system, the evolution problem involved in the JE can be solved. As a consequence, …
The special trajectory ensemble average (TEA), denoted by a subscript 0, in the Jarzynski Equality (JE) results in the Jensen inequality <R>_0 GT-EQ delta(F) for the work R done on the system, and not the thermodynamic work inequality <R> GT-EQ delta(F) since we find <R> NEQ <R>_0. Therefore, contrary to the common belief, the Jensen inequality does not directly support the JE as a nonequilibrium result. Jarzynski's microscopic treatment of the inclusive energy considers only the external work d_eE_k but neglects the ubiquitous change d_iE_k due to external-internal force imbalance, though d_i…
The Jarzynski equality (JE) is analyzed in regard to its validity for both quasi-static transformations in the thermodynamic limit and Hamiltonian evolutions of the work protocol. In the first case, we show that the JE holds for isothermal transformations only; in the second case, we show that the work done (linked to the final state Hamiltonian) depends on the temporal dynamics of the work protocol (including its speed), thus precluding the possibility of identifying it with the free energy or any other thermodynamic state function. Even in the case of thermodynamic limit and infinitesimally …
In the presence of feedback control by "Maxwell's demon," the second law of thermodynamics and the nonequilibrium equalities such as the Jarzynski equality need to be generalized. In this paper, we derive the generalized Jarzynski equalities for classical Hamiltonian dynamics based on the Liouville's theorem, which is the same approach as the original proof of the Jarzynski equality [Phys. Rev. Lett. 78, 2690 (1997)]. The obtained equalities lead to the generalizations of the second law of thermodynamics for the Hamiltonian systems in the presence of feedback control.
A short reply to Jarzynski's comment cond-mat/0509344 on my paper ``Flaw of Jarzynski's equality when applied to systems with several degrees of freedom'' (cond--mat/0508721) is presented.
We discuss and qualify a previously unnoticed connection between two different phenomena in the physics of nanoscale friction, general in nature and also met in experiments including sliding emu- lations in optical lattices, and protein force spectroscopy. The first is thermolubricity, designating the condition in which a dry nanosized slider can at sufficiently high temperature and low velocity exhibit very small viscous friction f ~ v despite strong corrugations that would commonly imply hard mechanical stick-slip f ~ log(v). The second, apparently unrelated phenomenon present in externally …
Responses of small open oscillator systems to applied external forces have been studied with the use of an exactly solvable classical Caldeira-Leggett (CL) model in which a harmonic oscillator (system) is coupled to finite $N$-body oscillators (bath) with an identical frequency ($ω_n=ω_o$ for $n=1$ to $N$). We have derived exact expressions for positions, momenta and energy of the system in nonequilibrium states and for work performed by applied forces. Detailed study has been made on an analytical method for canonical averages of physical quantities over the initial equilibrium state, which i…
We derive a general information-theoretic equality for a system undergoing two projective measurements separated by a general temporal evolution. The equality implies the non-negativity of the mutual information between the measurement outcomes of the earlier and later projective measurements. We show that it also contains the Jarzynski relation between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. Our result elucidates the information-theoretic underpinning of thermodynamics and explains why the Jarzynski r…
We present a method for detecting the entanglement of a state using non-equilibrium processes. A comparison of relative entropies allows us to construct an entanglement witness. The relative entropy can further be related to the quantum Jarzynski equality, allowing non-equilibrium work to be used in entanglement detection. To exemplify our results, we consider two different spin chains.
The past two decades witnessed important developments in the field of non-equilibrium statistical mechanics. Among these developments, the Jarzynski equality, being a milestone following the landmark work of Clausius and Kelvin, stands out. The Jarzynski equality relates the free energy difference between two equilibrium states and the work done on the system through far from equilibrium processes. While experimental tests of the equality have been performed in classical regime, the verification of the quantum Jarzynski equality has not yet been fully demonstrated due to experimental challenge…
We extend the Jarzynski equality, which is an exact identity between the equilibrium and nonequilibrium averages, to be useful to compute the value of the entropy difference by changing the Hamiltonian. To derive our result, we introduce artificial dynamics where the instantaneous value of the energy can be arbitrarily controlled during a nonequilibrium process. We establish an exact identity on such a process corresponding to the so-called Jarzynski equality. It is suggested that our formulation is valuable in a practical application as in optimization problems.
We show that the quantum Jarzynski equality generalizes to $\mathcal{PT}$-symmetric quantum mechanics with unbroken $\mathcal{PT}$-symmetry. In the regime of broken $\mathcal{PT}$-symmetry the Jarzynski equality does not hold as also the $\mathcal{CPT}$-norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system -- two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken $\mathcal{PT}$-symmetry is thermodynamically inhibited as the irreversible work diverges at the cri…
We discuss the theoretical foundations of non-equilibrium Monte Carlo simulations based on Jarzynski's equality and present, as an example of application, the determination of the running coupling in the Schrödinger-functional scheme.
The theory of phenomenological Non-equilibrium Thermodynamics is extended by includimg stochastic processes in order to account for recently derived thermodynamical relations such as the Jarzynski equality. Four phenomenological axioms are postulated resulting in a phenomenological interpretation of Jarzynski's equality. Especially, considering the class of Jarzynski processes Jarzynski's equality follows from the axiom that the statistical average of the exponential work is protocol independent.
We illustrate the Jarzynski equality on the exactly solvable model of a one-dimensional ideal gas in uniform expansion or compression. The analytical results for the probability density $P(W)$ of the work $W$ performed by the gas are compared with the results of molecular dynamics simulations for a two-dimensional dilute gas of hard spheres.
We have studied the Jarzynski equality (JE) in van der Pol and Rayleigh oscillators which are typical deterministic non-Hamiltonian models, but not expected to rigorously satisfy the JE because they are not microscopically reversible. Our simulations that calculate the contribution to the work $W$ of an applied ramp force with a duration $τ$ show that the JE approximately holds for a fairly wide range of $τ$ including $τ\rightarrow 0 $ and $τ\rightarrow \infty$, except for $τ\sim T$ where $T$ denotes the period of relaxation oscillations in the limit cycle. The work distribution function (WDF)…
The well-known Jarzynski equality, often written in the form $e^{-βΔF}=\langle e^{-βW}\rangle$, provides a non-equilibrium means to measure the free energy difference $ΔF$ of a system at the same inverse temperature $β$ based on an ensemble average of non-equilibrium work $W$. The accuracy of Jarzynski's measurement scheme was known to be determined by the variance of exponential work, denoted as ${\rm var}\left(e^{-βW}\right)$. However, it was recently found that ${\rm var}\left(e^{-βW}\right)$ can systematically diverge in both classical and quantum cases. Such divergence will necessarily po…
Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are app…
We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the se…
We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system $< x >$ to applied sinusoidal and step forces significantly depend on model parameters of mag…
