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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Traditional answers to what the 2nd Law is are well known. Some are based on the microstate of a system wandering rapidly through all accessible phase space, while others are based on the idea of a system occupying an initial multitude of states due to the inevitable imperfections of measurements that then effectively, in a coarse grained manner, grow in time (mixing). What has emerged are two somewhat less traditional approaches from which it is said that the 2nd Law emerges, namely, that of the theory of quantum open systems and that of the theory of typicality. These are the two principal approaches, which form the basis of what today has come to be called quantum thermodynamics. However, their dynamics remains strictly linear and unitary, and, as a number of recent publications have emphasized, “testing the unitary propagation of pure states alone cannot rule out a nonlinear propagation of mixtures”. Thus, a non-traditional approach to capturing such a propagation would be one which complements the postulates of QM by the 2nd Law of thermodynamics, resulting in a possibly meaningful, nonlinear dynamics. An unorthodox approach, which does just that, is intrinsic quantum thermodynamics and its mathematical framework, steepest-entropy-ascent quantum thermodynamics. The latter has evolved into an effective tool for modeling the dynamics of reactive and non-reactive systems at atomistic scales. It is the usefulness of this framework in the context of quantum thermodynamics as well as the theory of typicality which are discussed here in some detail. A brief discussion of some other trends such as those related to work, work extraction, and fluctuation theorems is also presented.

New experimental evidence (e.g., [

Of course, as is well known, thermodynamics preceded QM, and consistency with thermodynamics led to Planck’s law and the dawn of quantum theory. Following the ideas of Planck on black body radiation, Einstein in 1905 quantized the electromagnetic field [

States and observables: A physical observable has a one-to-one correspondence with a Hermitian operator

Dynamics: The dynamics is generated by the Hamiltonian operator

Collapse of the wave packet: Measurement generates an irreversible change of state to the system depending on the observable being measured.

The theory was criticized by Schrödinger and de Broglie both claiming that the state or wave function has a direct observable physical meaning. Moreover, de Broglie tried to formulate a complete alternative theory based on a nonlinear evolution equation. Einstein criticized the probabilistic interpretation and the non-local character of the theory as, in his words, violating “objective reality” as summarized in the EPR paper [

The measurement axiom has drawn significant criticism. One objection concerns the abrupt collapse of the wave function leading to irreversible change. This idea is counter to our intuition of differential changes. Schrödinger by introducing his famous cats criticized both the probabilistic interpretation and the objectivity of the theory, which seems to depend on the back action of either the cat or the human observer. Another remedy is the many world interpretation of QM due to Everett [

Nonetheless, quantum measurement theory has developed within the orthodox framework of QM with the purpose of streamlining the interpretation by replacing the abrupt collapse of the wave function by a more gentle differentiable dynamical evolution. What has emerged are two somewhat less traditional approaches from which it is said that the 2^{nd} Law emerges, namely, that of the theory of quantum open systems [

In contrast, the idea underlying the typicality approach [

These are the two principal approaches, which form the basis of what today has come to be called quantum thermodynamics. Both are firmly founded on the orthodox belief that the unitary dynamics of QM for all systems is foundational,

It should also be noted that there are a number of ways to prepare an ensemble described by a mixed density operator. Each appears to be physically very different from the other. For example, there is the heterogeneous ensemble representing a statistical mixture of homogeneous orthogonal state vectors introduced by Schrödinger [

Now, another useful mathematical framework within the quantum thermodynamic formalism is that which formulates the probability density distribution of an atomistic level system in terms of position and momentum and its momentum spectrum as discrete [

It is the first of these two mathematical modelling frameworks,

In the second fundamental orthodox approach, the theory of quantum open systems [

Finally, a description of the developed mathematical framework of SEA-QT and its applications (e.g., [

A number of publications [

In the paper at hand, we consider both the typicality of observables and of reduced states. The fashion according to which we “draw” our states from Hilbert space is neither just given by a restriction onto a projective subspace as in [

_{α}_{α}

In the following, we are interested in the Hilbert Space Average (HA) of an expectation value of an arbitrary Hermitian operator

According to the partitioning in subspaces α, the general operator

Using this decomposition, the HA yields:

where
_{α}

So far, however, this does not classify an 〈

Again, using the decomposition given in

This is a rigid result and my in principle be evaluated for any given AR

where

with _{α(β)} ≡ _{α(β)}/

In which cases can this be expected? Consider as an instructive example the case of no restriction, _{E}_{E}

The same overall picture can be considered more or less appropriate even if there are restrictions to different subspaces

Thus, measuring only one (or a few) observable(s) ^{2}(^{†}}.Thus, the distance between two operators

For the situation discussed above this HV is easy to calculate since

In the following, we analyze whether or not there is a typical state for a considered system which is in contact with some environment

To analyze typicality in this case, it is more convenient to write

The quantities _{lm}_{lm}_{lm}

Writing it this way we can use _{lm}_{S}

for this situation. Hence, whenever the dimension of the subspace onto which the full system is confined is much larger than the dimension of the Hilbert space of the considered system,

So far everything stated above has referred to the relative frequency of states featuring certain typical properties in Hilbert space. However, this does not imply directly anything rigorous on the dynamics, especially not on whether they may be classified as equilibrating, thermalizing,

with |_{m}_{n}_{m}_{n}

For a thermodynamic system one would expect the expectation value of a relevant variable to reach an equilibrium value that is independent of the details of the initial state, even though it may depend, e.g., on the overall energy,

So far we have been concerned with the question of whether or not certain QEV’s (or reduced states) will reach constant values that are independent of the details of the initial states, possibly after a very long time. The experiences with non-equilibrium thermodynamics are, however, even more far reaching. Thermodynamic observables not only reach final equilibrium values that are independent of the details of the initial states, but starting from the same non-equilibrium values, all evolutions, at any point in time, will be more or less the same irrespective of the details of the initial state. In the following, we turn towards the question of how this can be understood on the basis of typicality. More specifically we demonstrate that pure states from a set {|φ} featuring a common QEV of some observable

We present some analytical derivations based on the above considerations, in particular on

We specify our considered observable _{i}_{i}^{i}_{1} = 0, and normalized to _{2} = 1. Furthermore, we require the _{i}

For the following calculations we further introduce some kind of “substitute” ensemble {|ω＞}, which is much easier to handle. As will be shown below, this ensemble approximates the exact ensemble {|

The ensemble {|ω＞} is generated by:

where |

The construction

(of course, the average on the left-hand side corresponds to the substitute ensemble {|ω＞} while the average on the right-hand side is based on the completely uniform ensem {|

To assure that the ensemble {|ω＞} indeed approximates the ensemble {|

where

Of course, the states |ω＞ are not exactly normalized which would render them unphysical. However, one finds from

By exploiting

As defined above, the _{i}

The average of the QEV’s of

That is, the mean QEV can be adjusted through the choice of the parameter

The evaluation of HV[〈

In particular, one obtains Tr{^{2}}. Evaluating HV[〈

Again, since the _{i}

First, if one evaluates

Second, the upper bound from _{ω}

The mean QEV,

The HA[|

For ensembles close to equilibrium,

From this point of view some consequences on the applicability of projection operator theories (NZ, time-convolutionless, Mori formalism,

Nevertheless, to rephrase, the results of this paper indicate that in the limit of large (high-dimensional) systems, the inhomogeneity should become more and more irrelevant in the sense that the statistical weight of initial states, which yield an inhomogeneity that substantially changes the solution of the projected equation of motion, should decrease to zero. Note that this does not contradict the concrete results of [

The above results also shed some light on the relation of the apparently irreversible dynamics of QEVs to the, in some sense, reversible dynamics of the underlying Schrödinger equation. If a mean QEV as generated by some initial non-equilibrium ensemble (pertinent density matrix) relaxes to equilibrium (which can often be reliably shown [

The relaxations of QEVs from some initial non-equilibrium state to that of (stable) equilibrium (or possibly some other stationary state: unstable equilibrium, metastable equilibrium, or steady state) is of great interest for a large class of both reactive and non-reactive systems, particularly at the atomistic level. To predict these relaxations, one might use (at least in some cases) the reduced density operator and one of the equations of motion of the theory of quantum open systems [

The equation of motion of the SEA-QT framework governs how the diagonal and off-diagonal elements of the thermodynamic state or density operator (or matrix)

Where _{B}

Here

The

with

For a closed composite system composed of two distinguishable particles, assemblies of particles, fields, or a combination of these,

and

Finally, for a system experiencing a non-work interaction (

where the last term on the right accounts for either a heat or mass interaction. If the latter,

where

and the

The angle of rotation

The quantity

We now turn to a brief discussion of the application of each one of these equations of motion and a comparison of the results generated with experimental data found in the literature. The results presented and discussed are taken from [

Since the SEA-QT equation of motion implements the principle of SEA, its application to chemical kinetics is consistent with the idea put forward by Ziegler [

The SEA-QT kinematic models, which establish the energy and particle number eigenstructures for the finite- and infinite-eigenlevel, chemically reactive systems considered below, are not repeated here due to their complexity; and, thus, the reader is referred to [

For purposes of the comparisons given below, both the finite- and infinite-eigenlevel systems initially consist of 1 particle of _{2} contained in a tank with dimensions 4 nm × 2 nm × 2 nm and are governed by the following reaction mechanism:

An initial non-equilibrium state is established by finding a metastable equilibrium state far from equilibrium, which is then perturbed into the initial non-equilibrium state used by the equation of motion, _{f}_{b}_{f}_{b}_{f}_{b}^{−11} s and 15.027 × 10^{−11} s and are based on a fit of the SEA-QT results to the value of _{f}

This table also includes the values of _{f}

Finally, additional validation of the SEA-QT predictions is needed via a comparison of the forward reaction rate constants predicted with SEA-QT to those given in

In [

The SEA-QT model of the composite system considered here is that given in [_{B}_{1} supplied by source S′. This creates a state in a superposition of circular Rydberg levels |_{i}^{iϕ}^{−}^{iϕ}

After leaving the cavity, the atom is subjected again to a resonant microwave pulse in R_{2} equal to that at R_{1}, mixing the atom energy levels and creating a “

To measure the decay of coherence left on the field mode state by the Rb atom, a second atom of identical characteristics to the first is sent along the same path after a delay time of _{d.}

The red triangles with error bars correspond to the experimental values, while the blue line corresponds to the theoretical prediction made using the correlation functional of [

The SEA-QT prediction is given by the green line, which is the norm ‖_{A}_{B}_{A}_{B}

In [

The SEA-QT model for this system is that given in [

Both Turchette ^{2}〉 is used to represent the relaxation time.

Results for the SEA-QT simulations are compared with the experimental probability distribution _{G}

Comparisons with the experimental data shown in

The introduction of nonlinear terms into the von Neumann framework is, of course, not new nor necessarily original to IQT and its SEA-QT mathematical framework. Non-linear modifications of the von Neumann equation of motion have long appeared in the literature and are the result of two essentially distinct approaches, one fundamental and the other emergent [

In contrast to the above approach, the second approach tries to rationalize irreversibility by assuming that it is not an emergent effect of coarse graining and decoherence approximations but instead a fundamental phenomenon. This idea can be traced back to some of the work by Prigogine and collaborators [

Of course, it should be noted that a general criticism of fundamental non-linearity is also made by Peres [

Finally, laying the issue of emergent

Neither the typicality approach nor the eigenstate thermalization hypothesis nor the SEA-QT framework refer to “work” and “heat” as basic notions, although the latter is able to deal with these phenomenologically by treating them as work or heat interactions (e.g., see Section 3.3 above) between the system under consideration and some reservoir(s). A similar treatment of mass interactions is made within this framework [

which follows directly from the von Neumann equation. Here

which is a direct consequence, _{tot}_{S}_{R}_{SR}_{SR}_{S}_{SR}

A challenge for approaches based on

In order to overcome the aforementioned obstacles arising from

An approach similar to that of maximum work extraction but with a focus in the opposite direction is the starting point for so-called quantum fluctuation relations. The approach asks the following question: Given that a system is driven from one state to another, possibly far from equilibrium but in a prescribed fashion, what is the amount of work needed to drive this process? If the system is small, fluctuations may play a significant role and the amount of work becomes a stochastic variable. In this case, one asks for the work distribution or specific features of it. This approach also starts from

Here 〈…〉_{λ}

This paper has examined in some detail the theory of typicality, which is one of the two principal orthodox approaches that form the basis of what today is called quantum thermodynamics. The other, only very briefly described, is that of the theory of quantum open systems, which can be shown to emerge as a special case of the former. This second approach relies on a partition between the primary system and the environment and assumes the total state evolution of the composite system-environment to be unitary and generated by the composite Hamiltonian. The evolution of system state alone is then based on a reduced dynamics, the differential form for which is found via a Markovian master equation of the Lindblad type. The underlying “irreversible” dynamics, nonetheless, remains unitary. The detailed discussions surrounding the theory of typicality shed light on this apparent contradiction, namely, on the relationship of the so-called irreversible dynamics of QEVs to the reversible dynamics of the underlying Schrödinger equation. If, for example, a mean QEV as generated by some initial non-equilibrium ensemble relaxes to stable equilibrium, then for the majority of the individual states that form the ensemble, the corresponding individual QEVs will relax to stable equilibrium in the same way. Of course, there may be individual initial states giving rise to QEV evolutions that do not relax to stable equilibrium. Furthermore, inherent to this theory is the fact that it cannot directly predict the thermalizing dynamics of specific initial states as generated by specific Hamiltonians, since it primarily addresses relative frequencies.

To make such predictions, an unorthodox approach such as IQT and its mathematical framework SEA-QT can be used. The power of this rather unique approach to do so has been illustrated here via a number of applications of the SEA-QT framework to non-reactive and reactive systems. Validations of this framework via comparisons of predicted results to experimental and numerical data found in the literature demonstrate the power of this approach and support the claim that this framework provides an alternative, comprehensive, and reasonable explanation of irreversible phenomena at atomistic scales.

Finally, a brief discussion outlining the differences between emergent as opposed to fundamental non-linear approaches and the principal issues involved is given. This is followed by a presentation of some other recent trends in quantum thermodynamics, which involve approaches that are based on the thermodynamic notions of “work” and “heat”. The starting point for these approaches, which strive to arrive at a stringent definition of work, is an energy balance which follows directly from the von Neumann equation. The problem is that the balance is ill-defined. To address this, an approach based on the notion of a “weight”, which is usually a quantum system that has no degeneracies and a constant density of states, is used. This approach results in a formulation for the maximum extraction of work. A competing approach is that based on quantum fluctuations, which requires the assumption of a microreversible dynamics and results in a stochastic description for the amount of work extracted. Both approaches are shown to be equivalent in terms of the results obtained when the work distribution of the second approach is singular.

The first author would like to thank Christian Bartsch, Mathias Michel, Sheldon Goldstein, Roderich Tumulka, and Joel Lebowitz for fruitful discussions and collaborations on the subject of typicality and quantum thermodynamics, while the second author would like to thank Gian Paolo Beretta for their continuing collaboration on the generalization of thermodynamics to all scales of analysis. He also gratefully acknowledges the support provided by the U.S. Office of Naval Research under ONR grant N00014-11-1-0266 for a portion of the research results presented in this paper. In addition, both authors would like to thank Gian Paolo Beretta for organizing JETC13 at the Università di Brescia and for inviting us to be members of Panel K on

The introduction and conclusions (Sections 1 and 6) are a joint effort of both authors. The section on typicality (Section 2) is the work of the second author, while that on the SEA-QT framework (Section 3) is the work of the first author. The section on emergent vs fundamental non-linearity (Section 4) is a contribution from the first author and that on work and fluctuations (Section 5) primarily from the second author.

The authors declare no conflicts of interest.

^{9}Be+ ground state

^{21}Ne and a test of linearity of quantum mechanic

^{201}Hg

_{2}→HF+H

_{2})

_{2}→H

_{2}+H

_{2}→ HF + F Reaction”

_{2}and F+D

_{2}reactions

_{2}and fluorine atom + hydrogen/deuterium over the temperature range 240–373 K

_{2}and F+D

_{2}at T=295–765 K

_{2}/D

_{2}reactions on a new potential energy surface

Evolution of expectation value of the particle number for (a) the finite energy eigenlevel system and (b) the infinite energy eigenlevel system.

Reaction rate constant for initial temperature at 298 K for (

Chemical reaction rate for initial temperature at 298 K for (

Schematic representation of an atom-field Cavity QED experiment [

Comparison of the loss of coherence predicted by IQT/SEA-QT [

Comparison of the experimentally measured dissipative decay of cat state 3 with the SEA-QT predictions [

Non-equilibrium evolutions in thermodynamic state for the lowest 5 energy eigenlevels as well as for the lowest 100 [

Degrees of freedom for each of the molecules and atoms in the finite SEA-QT model [

Species | Translational Quantum #’s^{a} |
Vibrational Quantum #’s | Rotational Quantum #’s |
---|---|---|---|

1,…,400 | |||

_{2} |
1,…,400 | 0 | 0,1 |

1,…,400 | 0,1,2,3 | 0,1,…,7 | |

1,…,400 |

Although the translational principal quantum number

Values of the forward reaction rate constant reported in the literature for the reaction mechanism of

_{f}^{−11} (cm^{3}/molecule-sec) | |||||
---|---|---|---|---|---|

WH^{a} |
SBA^{b} |
HBGM^{c} |
RHPB^{d} |
WTM^{e} | |

298 | 2.33 | 2.48 | 2.93 | 2.81 | 2.26 |

Wurzberg and Houston [

Stevens, Brune, and Anderson [

Heidner, Bott, Gardner, and Melzer [

Rosenman, Hochman-Kowal, Persky, and Baer [

Wang, Thompson and Miller [