1. Maxwell's Demon
Suppose we have a box divided into two compartments. Each compartment contains a gas. The gases are not at the same temperature. There is a small hole in the wall between the compartments and a massless shutter that can be moved over the hole. When the shutter is over the hole, the two compartments are thermally isolated.
Now suppose that the shutter is operated by a very observant demon. The demon follows the following procedure: normally, he leaves the shutter shut, so that the two gases remain isolated. But when he sees a molecule from the cooler gas whose speed is greater than the average speed of the molecules in the hotter gas coming toward the shutter, he opens the shutter to let the molecule through. Similarly, he opens the shutter if he sees a molecule from the hotter gas whose speed is less than the average speed of the molecules in the cooler gas coming toward the shutter. (If the speed of the molecules is distributed according to the Maxwell-Boltzmann distribution, then there will be some such molecules.)
The effect of this will be to raise the temperature of the hotter gas and lower the temperature of the cooler gas. Since it seems that the demon need not perform any work, we apparently have here a process whose sole thermodynamic effect is a transfer of heat from a cooler to a hotter body, in violation of the second law of thermodynamics.Footnote 1
Maxwell first summoned his demon in 1867 (in a letter to Peter Tait; see Earman and Norton Reference Earman and Norton1998, 438). He took it to show that the second law is not, strictly speaking, true but only effectively true. This result may have seemed surprising at the time, but in the light of Loschmidt's reversibility objection to Boltzmann's H-theorem (Loschmidt Reference Loschmidt1876-77) and Poincaré's recurrence theorem (Reference Poincaré1890), this result is now generally accepted (see, e.g., Albert [Reference Albert2000], chap. 4, for a contemporary introduction to Loschmidt's reversibility objection and Poincaré's recurrence theorem).
Nonetheless, there is still a huge debate as to whether Maxwell's demon can operate in a way that achieves a net decrease of entropy. There seem to be two reasons for this. First, Maxwell's thought experiment seems to offer the prospect that we might, with sufficient ingenuity, be able to contrive net entropy decreases ourselves.Footnote 2 This would of course be tremendously useful. In fact, it sounds too good to be true. Second, there is an argument in the framework of the Boltzmann approach to statistical mechanics that Liouville's theorem prohibits Maxwell's demon from achieving a net decrease of entropy.Footnote 3 This second reason is the focus of this article.
2. Three Boltzmann Entropies
As noted, there is an argument in the framework of the Boltzmann approach to statistical mechanics that Liouville's theorem prohibits Maxwell's demon from achieving a net decrease of entropy. The entropy that is referred to in this argument is the Boltzmann entropy, SB, which is defined as follows:
In this equation, k is the Boltzmann constant. However, there is an ambiguity as to what exactly the Boltzmann entropy is because there are at least three interpretations of W in the literature: (i) W is the volume of phase space that is compatible with the macrostate of the system (e.g., Jaynes Reference Jaynes and Rosenkrantz1965/1983; Callender Reference Callender and Zalta2006; Hemmo and Shenker Reference Hemmo and Shenker2006), (ii) W is the number of arrangements compatible with the distribution of the system (e.g., Albert Reference Albert2000; Ainsworth Reference Ainsworth2005), and (iii) W is the probability of the macrostate of the system (e.g., Swendsen Reference Swendsen2008). It will be argued in the next three subsections that these three interpretations of W are not, in general, equivalent (although they are equivalent in special cases). Consequently, they give rise to three different Boltzmann entropies. In this article, these will be referred to as SB1, SB2, and SB3, respectively. For many purposes it may not be necessary to decide which of these entropies is intended. But, as we shall see in sections 3 and 4, the type of restriction(s) (if any) that Liouville's theorem places on Maxwell's demon's ability to achieve a net decrease of entropy does depend on which entropy one has in mind.
2.1. W as the Volume of Phase Space That Is Compatible with the Macrostate of the System
The reference to “phase space” in this interpretation of W refers to the γ-space of the system. For an N particle system (where each particle has 3 degrees of freedom), this space has 6N dimensions (three for the x, y, and z coordinates of each particle and three for the x, y, and z components of the velocity of each particle). The microstate of the system is represented by a single point in γ-space. We can partition the γ-space into regions, such that each region contains all and only those points that correspond to microstates that are compatible with a given macrostate of the system, where each macrostate corresponds to a unique set of values for some selected macroscopic properties (e.g., temperature, pressure, volume). On this interpretation of W, the value of W for a system is the volume of the region of γ-space that contains all the points that correspond to microstates that are compatible with the actual macrostate of the system.
This interpretation of W is probably the one that is the most commonly used, both in the physics literature and in the philosophy literature. For example, Jaynes asks us to “recall Boltzmann's original conception of entropy as measuring the logarithm of phase volume associated with a macroscopic state” (Reference Jaynes and Rosenkrantz1965/1983, 83), and Callender claims that “Boltzmann's great insight was to see that the thermodynamic entropy arguably ‘reduced’ to the volume in γ-space picked out by the macroscopic parameters of the system” (Reference Callender and Zalta2006).
2.2. W as the Number of Arrangements Compatible with the Distribution of the System
The easiest way to understand this interpretation of W is to consider the μ-space of the system. For an N particle system (where each particle has 3 degrees of freedom), this space has six dimensions (three for the x, y, and z coordinates of the particles and three for the x, y, and z components of the velocity of the particles). The microstate of the system is represented by N points in μ-space (where each point represents the position and velocity of a particular particle). Suppose that the μ-space is “coarse grained,” that is, partitioned into a number of equally sized cells of small but finite volume. The “distribution” of the system is given by specifying the number of particles in each cell. The “arrangement” of the system is given by specifying which particle is in which cell. On this interpretation of W, the value of W for a system is the number of arrangements that are compatible with the actual distribution of the system.
How does this interpretation of W compare to the first? Each arrangement of the system corresponds to a single coarse-grained cell in the system's γ-space. So each distribution corresponds to a number of such cells. So (in addition to the system's γ-space being divided into a number of coarse-grained cells: one for each possible arrangement) we can also think of it as being divided into a number of regions (one for each possible distribution), where each region contains all and only those cells that represent arrangements compatible with a particular distribution. So on this interpretation, the value of W for a system is essentially the volume of the region of γ-space that contains all the cells corresponding to the arrangements that are compatible with the actual distribution of the system (as well as being the number of arrangements that are compatible with the actual distribution of the system).
So there is a close link between these first two definitions of W. But the two definitions will only be equivalent if there is a one-to-one correspondence between the macrostates of the system and the distributions of the system, and, in general, this is not the case. For example, suppose we have an ideal gas (composed of identical molecules) whose temperature is not close to 0 K. And suppose we define the macrostate of the system by the values of the temperature, pressure, and volume of the system. The pressure, P, and temperature, T, are given by
In these equations, c1 and c2 are constants, N is the number of molecules in the gas, V is the volume of the gas, m is the mass of a single molecule of the gas, and 
is the mean-squared velocity of the molecules of the gas. There are several distinct distributions that can give rise to the same values of T, P, and V (and thus the same macrostate). Here, for example, are two:
i) A distribution in which the molecules are evenly distributed over volume V, and all have velocity of magnitude u.
ii) A distribution in which the molecules are evenly distributed over volume V, and half of the molecules have velocity of magnitude w, such that w2 = 2u2, and half have velocity 0.
So, in general, the distribution of the system is not the same as its macrostate. This point is sometimes ignored or glossed over in the literature, which might lead one to mistakenly conclude that the first two interpretations of W are equivalent. For example, Ainsworth sloppily asserts that the distribution “is effectively the macrostate of the system” (Reference Ainsworth2005, 631), and Albert sometimes uses the term “macrocondition” to refer to “distribution” (see, e.g., Reference Albert2000, 50), but he introduces the term “macrocondition” as follows: “there is patently a possible science of these temperatures and pressures and volumes—a science (that is) of macroconditions” (23), which seems to suggest that he takes macroconditions to be equivalent to (what are here called) “macrostates.”
2.3. W as the Probability of the Macrostate of the System
The third interpretation of W, as the probability of the macrostate of the system, is the least common. However, it has been urged by Swendsen as not only the interpretation that yields the most appropriate definition of entropy but also as the one that is closest to Boltzmann's own intentions: “The equation on Boltzmann's tombstone, S = klogW … does not refer to the logarithm of a volume in phase space. The equation was first written down by Max Planck, who correctly attributed the ideas behind it to Boltzmann [Planck Reference Planck1901, Reference Planck1906]. Planck also explicitly stated that the symbol ‘W’ stands for the German word ‘Wahrscheinlichkeit’ (which means probability) and refers to the probability of a macroscopic state” (Swendsen Reference Swendsen2008, 16). Swendsen himself argues for an epistemic approach to the probabilities involved here (17), suggesting that the probability of a macrostate is the degree of belief we have (or perhaps should have) that we will find the system in that macrostate, other things being equal. But it seems that one could also offer an ontological interpretation of the probabilities involved here. For example, a frequentist might suggest that the probability of a macrostate is the limit (as time goes to infinity) of the ratio of the length of time the system is in that macrostate to the length of time the system has existed.
How does this definition of W relate to the first?Footnote 4 Clearly, if the volume of γ-space that represents the macrostate of a system is proportional to the probability of that macrostate, then this definition of W is essentially the same as the first. Advocates of SB1 suppose this to be the case (see, e.g., Albert Reference Albert2000, 96). However, Swendsen claims that this supposition is not in general legitimate (so SB3 does not in general reduce to SB1). In particular, he claims that it is not legitimate if we are dealing with a system of strongly interacting particles.Footnote 5 Intuitively, this seems right: if, for example, there is a strong attractive force between the particles, then a macrostate in which the particles are all bunched together seems to be more likely than a macrostate in which the particles are spread out over a large volume, notwithstanding the fact that, other things being equal, the latter macrostate corresponds to a larger region of γ-space. Consider, for example, a cloud of gas that collapses to form a star. We know that the gas particles are more densely concentrated in coordinate space after the collapse. So, on the face it, we would expect the “star” macrostate to correspond to a smaller region of γ-space than the “dispersed cloud of gas” macrostate. But presumably the collapse is not an entropy-decreasing process. One way to make sense of what is going on here is to assign a nonuniform probability distribution over γ-space (and take entropy to be a function of probability, not γ-space volume). Relatively small regions of γ-space corresponding to gas all clumped together can then be judged to be more probable (higher-entropy) states than large regions corresponding to the gas widely dispersed.
However, Albert (Reference Albert2000, 90) suggests that we can account for what is going on in this case without resorting to a nonuniform probability distribution over γ-space. He suggests that the “star” macrostate will correspond to a larger region of γ-space than the “dispersed cloud of gas” macrostate because, although the particles in a star are more densely concentrated in coordinate space, they will be more widely dispersed in momentum space. But while it seems plausible that the particles will be somewhat more dispersed in momentum space (simply because the mean-square velocity of the particles will be higher), it is not obvious that this will be sufficient to compensate for the fact that the particles are more densely concentrated in coordinate space. Moreover, it is not clear that this kind of answer will be at all plausible in other cases where we have a system of interacting particles.
3. What Chains Does Liouville's Theorem Put on Maxwell's Demon's Ability to Reduce SB1?
As noted, in classical statistical mechanics the microstate of a system at a given time is represented by a point in the system's γ-space. In general, the microstate of the system will change over time. The evolution of its microstate over time can be represented by a line in the system's γ-space. As the system is governed by the laws of classical mechanics, and these laws are deterministic, the line does not branch. Moreover, we can, in principle, determine exactly what the line will look like, if we know exactly what the initial microstate of the system is.
But suppose that we do not know exactly what the initial microstate of the system is (as is of course generally the case). Suppose we only know that the system is initially in one of several possible microstates. We can represent our knowledge of the system's microstate by a region of its γ-space: the region that contains all the points corresponding to the microstates that, for all we know, the system might initially be in. Call that region A. We can use the laws of classical mechanics to work out what the possible microstates of the system are at some later time by “evolving” all the points in region A. This will give us a new region, B. In general, B will not have the same shape as A. But Liouville's theorem states that, if the system is isolated, the volumes of regions A and B will be the same.
3.1. Iron Chains?
Recall that SB1 is the Boltzmann entropy we get when we interpret W as the volume of γ-space that is compatible with the macrostate of the system. Liouville's theorem suggests an argument that Maxwell's demon cannot operate to decrease the SB1 of the gas without increasing his own SB1. The argument runs as follows: first, let's consider what happens to the gas, ignoring what happens to the demon. The gas begins in some macrostate GA and ends up in some macrostate of lower SB1, GB. Let's call the region of γ-space that contains all the points that represent microstates that are compatible with macrostate X “R(X).” As GB is a macrostate of lower SB1 than GA, R(GB) is smaller than R(GA). It seems that the initial microstate of the gas could be almost any of those that are compatible with GA, and the demon would still be able to operate. So the point representing the microstate of the gas could start off from nearly anywhere in R(GA) and would end up somewhere in R(GB). So—as R(GB) is smaller than R(GA)—the volume of the region that represents the possible microstates of the gas decreases (this is not forbidden by Liouville's theorem, as the gas is not an isolated system).
We know from Liouville's theorem that for an isolated system the volume of the region that represents the possible microstates of the system cannot decrease as the system evolves. The volume of the region that represents the possible microstates of a joint system is the product of the volumes of the regions that represent the possible microstates of the separate systems. By assumption, the gas + demon system is an isolated system. So if the volume of the region that represents the possible microstate of the gas decreases (and we saw above that it does), the volume of the region that represents the possible microstate of the demon must increase. So the volume of the region that represents the possible final microstates of the demon is greater than the volume of R(DA) (where DA is the initial macrostate of the demon). Suppose the final macrostate of the demon is DB. The region that represents the possible final microstates of the demon must be contained within R(DB).Footnote 6 So the volume of R(DB) must be greater than the volume of R(DA). So the SB1 of the demon must increase.Footnote 7
So it seems that Maxwell's demon can only successfully decrease the SB1 of the gas at the cost of increasing his own SB1. So it seems that Liouville's theorem prevents him from achieving a net decrease of SB1. Much of the literature on Maxwell's demon can be thought of as trying to give a physical explanation as to why the demon's SB1 must increase. The main suggestion is that the demon's SB1 must increase because to carry out his job the demon must gain information about the position and velocities of the molecules in the gas and that there is an SB1 cost associated with either (i) acquiring this information or (ii) erasing this information.Footnote 8 These claims have been criticized by Earman and Norton (Reference Earman and Norton1998, Reference Earman and Norton1999) and Norton (Reference Norton2005).
3.2. Aluminium Chains?
Albert (Reference Albert2000, chap. 5) and Hemmo and Shenker (Reference Hemmo and Shenker2006) point out that there is a flaw in the argument presented in the previous subsection.Footnote 9 As we have seen, Liouville's theorem, together with the fact that the region of γ-space that represents the final possible microstates of the gas is smaller than the region that represents the initial possible microstates of the gas, shows that the region of γ-space that represents the final possible microstates of the demon must be larger than the region that represents the initial possible microstates of the demon. But it is only if we assume that the demon has only one possible final macrostate that we can conclude that the whole of this region must be contained within a single macrostate. If the demon has several possible final macrostates: DB, DC, and so on, then all we can conclude is that the union of R(DB), R(DC), and so on, must contain the whole of the region representing the final possible microstates of the demon. And of course it is possible that R(DB), R(DC), and so on, are all individually the same size or even smaller than R(DA). So the SB1 of the demon might remain the same or even decrease.
This shows that the chains placed on Maxwell's demon by Liouville's theorem are not as restrictive as we had initially thought. But note that it still seems that the theorem places limitations on the operation of the demon. It seems to show that if the SB1 of the demon does not increase, then there must be some uncertainty as to what the final macrostate of the demon will be. (Although, Hemmo and Shenker [Reference Hemmo and Shenker2006] point out that in principle this uncertainty as to what the final macrostate of the demon will be can be translated into uncertainty as to what the final macrostate of the environment will be.)
3.3. Paper Chains ?
Let us distinguish two types of Maxwellian demon: “brazen demons” and “subtle demons.” A brazen demon wears its demonic powers on its sleeve, in the sense that everything that has the same macroscopic properties as a brazen demon is itself a (brazen) Maxwellian demon. Subtle demons do not wear their demonic powers on their sleeves, in the sense that not everything that has the same macroscopic properties as a subtle demon is itself a Maxwellian demon.
Intuitively, one might expect that any Maxwellian demon would be a subtle demon: if all we know about the (alleged) demon is really that he is in a certain macrostate (i.e., that he has certain values of temperature, pressure, volume, etc.), then it seems unlikely that we would know that he actually is capable of carrying out the operations required to reduce the SB1 of the gas. It seems that there could be lots of things that have the same macroscopic properties as the demon but cannot affect these operations. Or, to put it the other way round, if we know that the demon really is capable of carrying out the operations required to reduce the SB1 of the gas, then it seems that we know more about him than that he is in a certain macrostate. However, for present purposes there is no need to claim that a Maxwellian demon must be a subtle demon; it suffices to note that a Maxwellian demon could be a subtle demon.Footnote 10
Albert and Hemmo and Shenker do not draw the distinction between brazen and subtle demons and consider (in effect) only brazen demons. The central claim put forward in this article is that subtle demons are not subject to even the limitation suggested by Albert and Hemmo and Shenker: it is contended here that it is possible for a subtle demon to decrease the SB1 of the gas and end up (with certainty) in the same macrostate that he began in. This state of affairs is, intuitively, what the original thought experiment seems to suggest is possible. The operations that the demon is required to perform by Maxwell do not appear to be the sort of operations that would in any way alter the macrostate of the demon (as characterized by his macroscopic properties: temperature, pressure, etc.). So the claim of Albert and Hemmo and Shenker that there must necessarily be some uncertainty as to the final macrostate of the demon seems rather surprising (as does the claim they refute: that the final macrostate of the demon cannot be the same as his initial macrostate but must be a macrostate of higher SB1).Footnote 11
Consider a subtle demon (whose initial macrostate is DA). We may or may not know that he is a subtle demon (see n. 10). But either way, given that he really is a subtle demon, his possible initial microstates are actually only a subset of those that are represented by all the points in R(DA). From Liouville's theorem (together with the fact that the volume of γ-space that represents the final possible microstates of the gas is smaller than the volume of γ-space that represents the initial possible microstates of the gas), we know that the volume of γ-space that represents the final possible microstates of the demon is bigger than the volume of γ-space that represents the initial possible microstates of the demon. But as the initial possible microstates of the demon are actually only a subset of those that are represented by points in R(DA), this does not imply that the volume of γ-space that represents the final possible microstates of the demon cannot wholly lie in R(DA). So the final macrostate of the demon could be (with certainty) the same as his initial macrostate.
Call the subregion of R(DA) representing the microstates that the demon could initially be in (given that he is in macrostate DA and that he really is able to operate as a demon) A. Suppose the final macrostate of the demon will be (with certainty) the same as his initial macrostate (i.e., DA). We know by Liouville's theorem and the fact that the gas is in a lower-entropy state at the end of the experiment that the subregion of R(DA) representing the microstates that the demon could be in at the end of the experiment is larger than A. So at the end of the experiment, the “demon” might not be in A. If he is not, he will not be able to repeat his demonic machinations (i.e., he will no longer be a demon) since, by hypothesis, A contains all the microstates (in region R(DA)) from which he really is able to operate as a demon.
This is illustrated in figure 1 in which the dimensions representing the degrees of freedom of the gas are collapsed onto the y-axis, and the dimensions representing the degrees of freedom of the demon are collapsed onto the x-axis (cf. Hemmo and Shenker Reference Hemmo and Shenker2006). In figure 1A, the dashed lines enclose the region of phase space compatible with the initial macroscopic properties of the system (demon in state DA and gas in state GA). The solid lines enclose the region compatible with the macroscopic properties of the system and the fact that the demon is (initially) really a demon. In figure 1B, the dashed lines enclose the region of phase space compatible with the final macroscopic properties of the system (demon in state DA and gas in state GB). The solid lines enclose a region (compatible with the final macroscopic properties of the system) of the same volume as is enclosed by the solid lines in figure 1A. Evidently, there are some points in this region in which the “demon's” microstate does not lie in region A. If the system ends in one of these microstates, then the “demon” is no longer a genuine demon.
Figure 1. γ-space of the system at the start (A) and end (B) of the experiment.
Suppose, for the sake of definiteness, that the region representing the final possible microstates of the demon is three times as big as region A (as in fig. 1). Then, other things being equal, in the best case scenario, the chance that the demon will not be able to repeat the operation is 2/3. But suppose that the demon can repeat the operation (i.e., he does end up in region A). We know that, second time around, he might end up in a microstate that does not lie in A (because, once again, the volume of γ-space representing the possible microstates of the gas has decreased, so the volume of γ-space representing the possible microstate of the demon must increase). So, even if he can repeat the operation once, he might not be able to repeat it a second time. And the chance that he will be able to carry out the operation N times tends to zero as N tends to infinity: sooner or later the demon will not be able to operate.
So, although Liouville's theorem does not prohibit the existence of a Maxwellian demon that (i) can cause a net decrease of SB1 and (ii) is guaranteed to end up in the same macrostate that he began in, it does imply that if the demon fulfills these conditions then there is no guarantee that he will be able to repeat the operation, and the chance that he will be able to carry out the operation N times tends to zero as N tends to infinity.
4. What Chains Does Liouville's Theorem Put on Maxwell's Demon's Ability to Reduce SB2 and SB3?
Recall that SB2 is the Boltzmann entropy we get when we interpret W as the number of arrangements compatible with the distribution of the system, and SB3 is the Boltzmann entropy we get when we interpret W as the probability of the macrostate of the system. In this section, we consider how the arguments of the previous section are affected if one changes one's concern from the demon's ability to achieve a net decrease of SB1 to its ability to achieve a net decrease of SB2 or SB3.
How do things change if we substitute SB2 for SB1 in the foregoing arguments? The initial argument and the response of Albert and Hemmo and Shenker to this argument are essentially unchanged. However, the final argument, which purports to show that Liouville's theorem does not prohibit the existence of a Maxwellian demon that (i) can cause a net decrease of SB1 and (ii) is guaranteed to end up in the same macrostate that he began in, does not go through if we substitute SB2 for SB1. This argument relied on the supposition that there could be a subtle demon, that is, that there could be a demon that could only operate from a special subset of the microstates compatible with its initial macrostate. The analogue of a subtle demon in the case of SB2 would be a demon that could only operate from a special subset of the arrangements compatible with its initial distribution. But it seems that there could be no such demon: different arrangements compatible with a given distribution are permutations of one another, and it seems that if a demon is actually capable of carrying out the operations that Maxwell's demon carries out when he has one arrangement, then he should also be able to carry out those operations if he has a permutation of that arrangement. So, it seems that Liouville's theorem does prohibit the existence of a Maxwellian demon that (i) can cause a net decrease of SB2 and (ii) is guaranteed to end up with the same distribution that he began with.
If we substitute SB3 for SB1, then the argument does not even get off the ground. Liouville's theorem puts restrictions on how volumes of γ-space evolve. In the case of SB3, W is taken to be the probability of the macrostate of the system (and, in general, it is not assumed that there is a straightforward connection between probabilities and γ-space volumes). So, in general, Liouville's theorem puts no constraints on what happens to SB3.
In particular, if we are taking entropy to mean SB3, then the fact that the gas goes from a higher- to a lower-entropy state no longer means that the region representing the macrostate of gas at the end of the procedure is smaller than the region representing the macrostate of gas at the start of the procedure since W is now taken to be the probability of the macrostate of the system, not the volume of the region that represents it in γ-space. Moreover, as the volume of the region representing the possible microstates of the gas need not change over the course of the operation, the volume of the region representing the possible microstates of the demon need not change over the course of the operation. So, the region representing the final possible microstates of the demon could be exactly the same as the region representing the initial possible microstate of the demon. So, in this case the demon could continue to operate indefinitely.
However, if the demon operates on a gas of noninteracting particles (so the probability of a macrostate of the gas is essentially the same as the volume of the region that represents it in γ-space), the fact that the gas goes from a macrostate of higher SB3 to macrostate of lower SB3 does mean that the region representing the macrostate of the gas at the end of the procedure is smaller than the region representing the macrostate of gas at the start of the procedure after all. And this means that the region representing the final possible microstates of the demon must be larger than the region representing the initial possible microstates of the demon. So we have essentially the same results as in the case of SB1.
5. Summary of Results
The main results of the article are (1) Liouville's theorem does not prohibit the existence of a Maxwellian demon that (i) can cause a net decrease of SB1 and (ii) is guaranteed to end up in the same macrostate that he began in, but it does imply that (iii) if the demon fulfills these conditions then there is no guarantee that he will be able to repeat the operation, and the chance that he will be able to carry out the operation N times tends to zero as N tends to infinity. (2) Liouville's theorem does not prohibit the existence of a Maxwellian demon that (i) can cause a net decrease of SB2, but it does imply that (ii) if the demon fulfills this condition then there is no guarantee that he will end up with the same distribution that he began with. (3) Liouville's theorem does not prohibit the existence of a Maxwellian demon, operating on a gas of interacting particles, that (i) can cause a net decrease of SB3, (ii) is guaranteed to end up in the same macrostate that he began in, and (iii) can repeat his machinations indefinitely. (4) Liouville's theorem does not prohibit the existence of a Maxwellian demon, operating on a gas of noninteracting particles, that (i) can cause a net decrease of SB3 and (ii) is guaranteed to end up in the same macrostate that he began in, but it does imply that (iii) if the demon fulfills these conditions then there is no guarantee that he will be able to repeat the operation, and the chance that he will be able to carry out the operation N times tends to zero as N tends to infinity.
