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1. Introduction
Our everyday experiences tell us that tending toward and reaching equilibrium is ubiquitous. This is enshrined in the Second Law of thermodynamics. Classical statistical mechanics (CSM) aims to account for such macroscopic behavior in terms of classical dynamics governing the microscopic constituents and some probabilistic assumptions. While having received a lot of philosophical attention recently, the foundational project of CSM is still an open problem. Some would argue that progress is being made in this manner; others are sceptical: “Can we explain the laws of thermodynamics, in particular the irreversible increase of entropy[?] Attempts based on classical dynamics have all failed” (Hemmo and Shenker Reference Hemmo and Shenker2001, 562).
Quantum statistical mechanics (QSM) aims to succeed where CSM has purportedly failed: it aims to account for the relevant macroscopic behavior in terms of quantum dynamics governing the microscopic constituents and some probabilistic assumptions.Footnote 1 There have been several recent attempts in this vain. These are set out below.
In a series of papers, Albert (Reference Albert1994a, Reference Albert1994b, Reference Albert2000) argues that GRWian quantum mechanics entails equilibration. Here is an extremely rough sketch of Albert's strategy: Distinguish between two kinds of possible evolutions (trajectories) for a quantum system: ‘thermodynamically normal’ and ‘thermodynamically abnormal’. ‘Thermodynamically normal’ trajectories are those such that the entropy of the system increases over time, ‘thermodynamically abnormal’ are those where it decreases. Albert conjectures that there are orders of magnitude more ‘thermodynamically normal’ trajectories than ‘abnormal’ ones and that GRWian genuinely random collapse entails equilibration because the ubiquity of collapse makes it overwhelming likely that a system on an abnormal trajectory gets ‘knocked onto’ a thermodynamically normal trajectory. While this is an interesting proposal, Albert provides neither a rigorous statement nor defense of this conjecture.
Hemmo and Shenker (Reference Hemmo and Shenker2001, Reference Hemmo and Shenker2003) argue for a similar result. Their argument is structurally the same as Albert's, but instead of making recourse to GRWian quantum mechanics they use decoherence models. They propose the following conjecture: “perturbations of the molecules in the decoherence interaction are enough to put them with high probability on thermodynamic-normal trajectories” (Hemmo and Shenker Reference Hemmo and Shenker2001, 566). While also an interesting proposal, it rests, like Albert's, on conjecture.
A recent paper by Linden et al. (Reference Linden, Popescu, Short and Winter2008) purportedly goes beyond conjecture: “We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems” (4). If this proof is sound it would be a most startling result; to borrow a phrase of Callender's, it would constitute finding the Holy Grail of statistical mechanics. I call the approach that Linden et al. develop the ‘Aharonov approach’.Footnote 2
In section 2, I present Linden et al.'s argument that reaching equilibrium is a universal property of quantum systems. In section 3, I identify three kinds of challenges for the Aharonov approach. In section 3.1, I show that their argument is not persuasive as it stands. I offer a rehabilitation of their argument; this forms the bulk of the article. In section 3.2, I consider a broader challenge to the approach, namely, that it is untenable because it is interventionist. I defend the Aharonov approach against this charge. In section 3.3, I briefly discuss a key assumption for the theorems. In section 4, I present my conclusions.
2. The Aharonov Approach
How does the Aharonov approach purport to account for a particular system tending toward and remaining in equilibrium? One starts by considering a larger system—call this the ‘global system’. The particular system, that is, the system which is to be shown to tend toward equilibrium, is a smaller system within the global system. Call that the ‘subsystem’. The rest of the global system, that is, just that part of the global system that is not the subsystem, forms the environment or ‘bath’ of the subsystem—call this the ‘bath’. This is the (albeit somewhat abstract) ‘metaphysical picture’ of the Aharonov approach.
All the possible states of the global system are represented by a Hilbert space, 
. This is decomposed into the subsystem and the bath: 
, where 
and 
are the Hilbert spaces of the subsystem and the bath, respectively. The formal results hold for any arbitrary decomposition of 
.Footnote 3 However, an important physical assumption will be that the subsystem is significantly smaller than the global system and, therefore, the bath.
The global system—and, therefore, the subsystem and the bath—is posited to evolve according to a Hamiltonian, H. The Hamiltonian is “completely general” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 4) except for one constraint, namely, that it is assumed to have nondegenerate energy gaps. (I return to this assumption in sec. 3.3.) The main strategy of the Aharonov approach is to show that subsystems reach equilibrium in virtue of being embedded in global systems of some such dynamics.
To state and assess the central results of their paper, it is necessary to introduce some further technical machinery. One of the problems of their paper, as I see it, is that there is a considerable ‘conceptual gap’ between the formal results and the conclusions drawn from them. My aim in the rest of this section is just to present their results, and hence I follow their paper closely. In the next section, I critically assess all this.
Let 
denote the global pure state of the system at time t. This state can be represented as a density matrix 
. The density matrices 
and 
represent the state of the subsystem and the bath, respectively. They relate to one another by tracing over the global state in the following way: 
and 
.
Central to the Aharonov approach are time-averages, the time-averaged state of the system, target system, and bath. These are denoted by 
, 
, and 
, respectively and are defined as follows:
Two more items are needed. First, the “effective dimension,” 
, of a mixed state, 
:
The former “tell us, in a certain sense, how many pure states contribute appreciably to the mixture” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 3). As regards the latter, “The trace-distance characterizes how hard it is to distinguish two states experimentally (even given perfect measurement). When it is small, the two states are indistinguishable. More precisely, it is equal to the maximum difference in probability for any outcome of any measurement performed on the two states” (3).
Consider 
—the trace-distance between the state of the target system at a particular time and the time-averaged state of the subsystem. And now consider the time-average of this distance: 
. With this machinery one can state the key theorem of Linden et al. (Reference Linden, Popescu, Short and Winter2008, 4):
Theorem 1. For any state 
evolving under a Hamiltonian with nondegenerate energy gaps, H
What does this theorem tell us? “By bounding 
, [the] theorem tells us that the [subsystem] will equilibrate whenever the effective dimension explored by the bath 
is much larger than the [subsystem] 
, or whenever the effective dimension explored by the total state 
is much larger than two copies of the [subsystem]. … In other words, if 
is small, the [subsystem] spends most of its time close to the equilibrium state” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 4).
If this is persuasive then a further argument is needed to the effect that 
is small. Linden et al. (Reference Linden, Popescu, Short and Winter2008, 5) provide a second theorem to this end:
Theorem 2. (i) The average effective dimension 
where the average is computed over uniformly random pure states 
is such that
(ii) For a random state 
, the probability 
that 
is smaller than 
is exponentially small, namely,
where 
.
Here is how Linden et al. characterize the content of the second theorem: “Point (i) essentially tells us that the average effective dimension is larger than half the dimension of the Hilbert subspace [
], so when we draw states [at random] from a subspace of large dimension, the effective dimension 
of a typical state is large. Point (ii) makes the result even sharper, telling us that the probability of having a small effective dimension is exponentially small” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 5).
For convenience, abbreviate ‘
’ to ‘
’. Their argument can be summarized as follows: First, they state that “when 
is small, the subsystem equilibrates to 
” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 4). By theorem 1 
is bound below the ratio of twice the dimension of the target system, 
, to the effective dimension of the whole/global system, 
. When the dimension of the target system is far smaller than the global system, that is, when 
is far larger than 
, 
must be very small. Given that 
is small by hypothesis, what needs to be shown is that 
is large. The second theorem is intended to show this: it bounds 
above half the dimension of the global system (which is large by hypothesis). From this, suggest Linden et al., it follows that 
is typically small. Thus, they claim, a small subsystem of a large global system will tend toward, and then remains in, equilibrium.Footnote 4
3. Challenges for the Aharonov Approach
As it stands, the argument presented in the previous section is not compelling: the Aharonov approach faces a number of challenges, which need to be met if it is to deliver on its promises. These can be categorized into three kinds.Footnote 5 The first kind pertains to the conceptual framework of the approach in a narrow sense. In essence, this is the question of whether the two theorems presented do give the desired conclusion about quantum systems reaching equilibrium. A critical discussion of this forms section 3.1.
A second kind of problem has to do with the conceptual framework of the approach in a broader sense. In particular, the Aharonov approach is an interventionist one, but interventionism is considered to be a dubious stance in the context of the foundations of statistical mechanics. To make the Aharonov approach tenable, one needs to defend an interventionist stance. A critical discussion of this forms section 3.2.
A third kind of challenge concerns the assumption about the Hamiltonian. This is taken up briefly in section 3.3.
3.1. Narrow Conceptual Challenges
Recall the summary of Linden et al.'s argument given at the end of the section 2. Two things are amiss with the argument: First, there is no argument that 
being small is an adequate representation of the subsystem reaching and remaining at equilibrium. I take this up in section 3.1.1. Second, it is not clear, I suggest, how the results in the second theorem relate to 
. In particular, it is not clear how, if at all, theorem 2 shows that 
is typically small. I take this up in section 3.1.2. I then offer a defeasible argument for a desirable (though weaker) conclusion in section 3.1.3.
3.1.1. Representing Equilibrium
It is standard that density matrices represent states of the system and that the Hamiltonian governs its evolution. The state of the subsystem is also represented by density matrices on a restricted Hilbert space, and it too evolves according to the governing Hamiltonian.
The time-averaged state of the subsystem, 
, is obtained by integrating the density matrix representing the subsystem with respect to time and evaluating it in the time-limit, 
. This is a standard procedure, resulting in an uncontroversial mathematical object. But what does this represent physically? While it is not explicitly stated in the paper, 
is tacitly assumed to represent the equilibrium state. It is only by taking this to represent the equilibrium state that 
relates to equilibrium at all. By construction, it is the (abstract) state in which the macroproperties of the system do not change. It is important to note that this is a weak notion of equilibrium: it does not indicate the values of the macroproperties in this state, or the ‘stability’ of such a state, in the sense of whether it is impervious to small perturbations.Footnote 6 I assume that the weak notion of equilibrium here is sufficient, other challenges notwithstanding, for the current proposal to work.
Next the trace-distance is used to relate the state of the system at a particular time, 
, to the equilibrium state, 
. What does the trace-distance represent? Formally the trace-distance “is equal to the maximum difference in probability for any outcome of any measurement performed on the two states” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 5). The issue is how to understand this physically. In particular, how to substantiate the claim that “[it] characterizes how hard it is to distinguish two states” (5).
First note that the claim is probabilistic; conceptually, 
is not equivalent to the claim that the measurements performed on the two states will have the same outcome. Rather, 
says that the probability of getting different outcomes is zero.Footnote 7 The smaller the value of 
, the closer are the expectation values for any operator acting on each state; the more likely it is that the outcome of measurements performed on the two states will be the same.Footnote 8 Why is the probability of there being a difference in outcomes of measurements performed on states a suitable proxy for the physical (dis)similarity, or (in)distinguishability, of the states? The brief answer is that, owing to superpositions, quantum systems are in definite physical states only upon measurement.Footnote 9 It is in this sense that the trace-distance can be thought to characterize the (dis)similarity of two states. In the present case, then, 
is to be regarded as a measure of how physically similar the actual state of the subsystem at a particular time is to the equilibrium state. This is an important point which is worth restating: 
is a measure of the difference in the expectation values of operators on each of the two systems, which is the probability of getting different outcomes upon measurement for any given observable. This measure, I argued above, is a suitable proxy for the physical similarity between the actual state of the system at a particular time and the equilibrium state. Put somewhat colloquially, the smaller the value 
, the closer the subsystem is to equilibrium.
Now consider 
. Given the above, this represents how close the subsystem is to the equilibrium state on average. It follows that if 
is small, then the subsystem must spend most of its time close to equilibrium. However, making recourse to 
has its limitations. Like equations (1), (2), and (3), 
is the time-average in the limit, 
. 
being small is, therefore, compatible with the subsystem taking an arbitrarily long (but finite) amount of time to reach equilibrium. It is also compatible with the subsystem deviating arbitrarily far from equilibrium once it has reached it. Thus, a small value for 
cannot represent the subsystem monotonically tending toward, reaching, and remaining in equilibrium.
Does this show the Aharonov approach to have failed? In the context of CSM, Callender (Reference Callender2001) has argued that seeking an account of monotonic tending toward, reaching of, and remaining in equilibrium would amount to taking thermodynamics too seriously. Specifically, the Second Law of thermodynamics taken as an exceptionless law is false because real systems exhibit fluctuations away from equilibrium. In this sense, the Aharonov approach has not failed. What is really at stake is the character of the fluctuations. Fluctuations are small in real systems. Without an account of the possible size of fluctuations it may be argued that the Aharonov approach remains unsatisfactory. By making a stronger assumption about the Hamiltonian used in theorems 1 and 2, Linden et al. also provide probabilistic bounds on deviations away from equilibrium to the effect that large deviations are unlikely (cf. Linden et al. Reference Linden, Popescu, Short and Winter2008, 12). Whether this assumption about the Hamiltonian is in itself justifiable needs to be examined, and I do so in section 3.3. However, irrespective of this result, a small value for 
would show that the subsystem must on average be close to or in the equilibrium state and that deviations away from it can only be temporary. And this would be a significant result, I suggest, especially so because it follows from very general considerations of the dynamics governing the behavior of arbitrary subsystems of large quantum systems.
3.1.2. State-Averaging
Having analyzed the first theorem and the significance of 
being small, I now turn to the second theorem which purports to show that 
is typically small. Theorem 2 purportedly underpins the claim that 
is typically small. How so? By theorem 1, 
is small if 
is large (because 
is small by hypothesis). Recall what Linden et al. write about part i of theorem 2: “[It] tells us that the average effective dimension is larger than half the dimension of the Hilbert subspace [
], so when we draw states [at random] from a subspace of large dimension, the effective dimension 
of a typical state is large” (Reference Linden, Popescu, Short and Winter2008, 5; my emphasis).
Their idea is that by bounding 
in this way, a randomly drawn state from 
will typically have a large effective dimension (because 
is large by hypothesis). So any small subsystem of a typical random state will equilibrate because the effective dimension of the state will be large.
The crucial ingredient here is ‘state-averaging’; 
is calculated by taking the average of 
for all states 
by assigning a uniform distribution over 
(i.e., by giving each state the same ‘weight’). However, it is not the case that the property ascribed to the average of 
is also a property of a particular 
—not even typically. In theorem 2 an average state is ascribed a particular property by being bound to the dimensionality of the Hilbert space from which the states, which make up the average state, are drawn. Whether any particular state has this property cannot be inferred from the theorem, however. An example to illustrate: the average height of four people may be 170 cm, but it does not follow that any individual has a height of (anywhere close to) 170 cm. For all one can infer from the average alone, two of the people may be toddlers and two basketball players. In the present context it has not been demonstrated that (anything like) the overwhelming number of states have the relevant property, so a typicality argument does not even get off the ground.Footnote 10
The second part of theorem 2 is not redundant, however. Pace Linden et al., it is not the case that part ii “makes the result [in part i] even sharper” (Reference Linden, Popescu, Short and Winter2008, 5); rather part ii is a conceptually distinct claim. Specifically, it says that for any state 
, the probability that 
is less than 
is exponentially small. It follows from this that the probability that 
is greater than 
is exponentially large. In the next section, I show this can be used to rehabilitate the Aharonov approach.
3.1.3. Reaching Equilibrium
Given the clarifications and amendments in the previous section, one can now construct an argument for the conclusion that arbitrary small subsystems of large quantum systems typically tend toward and remain in, or close to, equilibrium.
1. As per section 3.1.1, if
is large relative to , then an arbitrary small subsystem of a large quantum system tends toward and remains in, or close to, equilibrium.2. As per section 3.1.2, for a randomly drawn state
, the probability that is greater than is exponentially large.3. By hypothesis,
is large relative to .4. From items 2 and 3,
is typically large relative to .5. From items 1 and 4, an arbitrary small subsystem of a large quantum system typically tends toward and remains in, or close to, equilibrium.
is large relative to 
, then an arbitrary small subsystem of a large quantum system tends toward and remains in, or close to, equilibrium.
, the probability that 
is greater than 
is exponentially large.
is large relative to 
.
is typically large relative to 
.Notice that on this account, the systems of interest (i.e., those represented by the subsystem) do not universally tend toward and remain in equilibrium for two distinct reasons. First, some systems may be atypical in the sense given above. While the probability of this is exponentially small, it is not ruled out. Second, and as detailed in section 3.1.1, even typical states can take arbitrarily long to reach equilibrium and can deviate away from it.
3.2. Interventionism
I now consider a ‘broader’ conceptual challenge. The Aharonov approach is clearly an interventionist approach: the system of interest is construed as a subsystem of a large quantum system, and it is in virtue of being embedded in this larger environment (or bath) that it equilibrates. However, interventionism is, insofar as it is discussed, considered to be a misguided stance in the context of the foundations of statistical mechanics (cf. Davies Reference Davies1974; Horwich Reference Horwich1987; Sklar Reference Sklar1993; Bricmont Reference Bricmont, Gross, Levitt and Lewis1997; Ridderbos and Redhead Reference Ridderbos and Redhead1998; Frigg Reference Frigg and Rickles2008).
One can identify three arguments in the literature for the conclusion that interventionism is misguided: “all for nothing,” “incredulity,” and “deus ex machina.” I consider each in turn and conclude that none of them is a knockdown argument against the Aharonov approach.
3.2.1. All for Nothing
The “all for nothing” argument is that an interventionist stance in the foundations of statistical mechanics is all for nothing: statistical mechanics aims to account for the Second Law of thermodynamics, but the Second Law only pertains to isolated systems. For example, Ainsworth writes: “There is a tension (with the second law) which makes claims only about the thermodynamic entropy of closed systems and a difficulty in spelling out a modified version of the law more precisely; after all, thermodynamic entropy does not “always increase in open systems (which is why the second law was originally formulated with respect to closed systems!)” (Ainsworth Reference Ainsworth2005, 627).
First, it is important to note that the current proposal does not purport to recover the Second Law of thermodynamics (i.e., the reductive relation between thermodynamics and QSM is not the concern of the article); rather, it aims to give an account of a phenomenon for which thermodynamics also (purportedly) accounts. So one cannot infer the failure of the Aharonov approach in virtue of it not recovering a law that only pertains to isolated systems.
The more substantial point is this, however: quantum mechanics tells us that entanglement is ubiquitous and that, therefore, there are no genuinely isolated systems after all. Of course, not all nonisolated systems do exhibit equilibration, but it need not be in virtue of their being nonisolated that they exhibit the behavior that they do. All of this is compatible with the Aharonov approach, as it does not make an exceptionless claim about the behavior of systems; it claims that systems typically equilibrate. A challenge would then be to show why some systems are atypical in this respect, but there is no reason to think that this challenge cannot be met within the framework itself.
3.2.2. Incredulity
The “incredulity” argument is sometimes heard in informal discussions of interventionism. The idea is that it is simply incredulous that the causal efficacy of the environment is a necessary condition for the exhibited behavior of the system in question. An example: “I cannot with a straight face tell a student that (part of) our explanation for irreversible phenomena on earth depends on the existence of Sirius” (Bricmont Reference Bricmont, Gross, Levitt and Lewis1997, 154).
This is not an argument, however; one cannot, in the words of Lewis, “refute an incredulous stare.” Bricmont's sentiment seems to be that an account of the behavior of systems ought not to depend on the environment in which the system is embedded. But the ‘ought’ is a stipulation, and the burden of a motivation for it lies squarely on the shoulders of the interventionist sceptic.
3.2.3. Deus Ex Machina
A more interesting argument against interventionism comes in the form of “deus ex machina.” As Frigg (Reference Frigg and Rickles2008, 164) puts it, “A common objection against [interventionism] points out that we are always free to consider a larger system, consisting of our original system and its environment. For instance, we can consider the gas cum box system … Interventionism … seems wrong because it treats the environment as a kind of deus ex machina that is somehow outside physics; but the environment is governed by the fundamental laws of physics just as the system itself is and so it cannot do the job that the interventionist has singled out for it do.”
Whether or not this is a persuasive argument against interventionism in general (or in CSM, the context in which Frigg is writing) is orthogonal to the current considerations. The kind of explanatory regress that is highlighted does not take place in the Aharonov approach because the environment is not treated as a kind of deus ex machina. Indeed, it is clear from section 3.1 that it is in virtue of the entire target system cum environment being governed by the same physical dynamics that the earlier results hold.
3.3. Hamiltonian
A selling point of the Aharonov approach as Linden et al. present it is that the results follow from some very general assumptions about quantum systems. Of particular interest is the assumption made about the governing Hamiltonian. They emphasize that “the restriction to Hamiltonians that have no degenerate energy gaps is an extremely natural and weak restriction” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 2).
In what sense is this an “extremely natural and weak restriction”? Linden et al. offer two defenses of it. First, in brackets directly after the above quote they write: “Indeed, adding an arbitrarily small random perturbation to any Hamiltonian will remove all degeneracies” (Reference Linden, Popescu, Short and Winter2008, 2). This is obviously an ad hoc move. More significantly, though, it falls prey to the “deus ex machina” argument outlined in section 3.2.3: the perturbation comes from “outside” of the system and is, in this sense, nonphysical. Second, in a discussion of the derivation of theorem 1, they write: “we did not assume anything special about the interaction (apart from not having degenerate energy gaps—which rules out only a set of Hamiltonians of measure zero)” (4).
Whether this is a justification of the restriction depends on one's response to the so-called measure zero problem (see Sklar Reference Sklar1993, 182–88). A lack of space precludes a detailed discussion of this problem.
One final remark about the Hamiltonian: in section 3.1.1, I noted that, by making a stronger assumption, Linden et al. provide a theorem to the effect that large deviations away from equilibrium are unlikely. Linden et al. characterize this stronger assumption as the quantum analog to classical ergodicity. Assuming that systems are ergodic is, of course, a controversial matter, and I do not engage with that debate here. It is just interesting to note that assuming ergodicity can provide powerful results in the context of QSM too.
4. Conclusion
The Aharonov approach purports to succeed where other approaches in QSM have failed, namely, by proving in almost full generality that “reaching equilibrium is a universal property of quantum systems” (Linden et al. Reference Linden, Popescu, Short and Winter2008, 4). In section 2, I detailed Linden et al.'s argument for this conclusion. In section 3, I outlined the various challenges the Aharonov approach faces. Specifically, in section 3.1, I argued that, as it stands, Linden et al.'s proof is not compelling. By providing conceptual clarifications and amendments, I constructed an argument for the conclusion that an arbitrary small subsystem of a large quantum system typically tends toward and remains in, or close to, equilibrium. This is the central result of the article. In section 3.2, I outlined a broader conceptual challenge for the Aharonov approach: its being interventionist. I identified three arguments against interventionism and defended the Aharonov approach against each. Finally, in section 3.3, I briefly discussed the assumptions made about the dynamics of systems.
Clearly more work needs to be done before we find the proverbial Holy Grail of statistical mechanics. However, the Aharonov approach does give us an exciting and promising new direction for exploration.
, are not states on which actual measurements can be performed. Note, also, that 
is not a measure of the difference in the actual outcomes.
