Michael te Vrugt

Consider a system of two classical particles that have all their qualitative properties (mass, shape, form, and so on) in common. Does it make a difference if you swop their positions? Despite seeming innocent, this question touches upon a very deep philosophical problem. If you say that it does (or at least that it can) make a difference, you are committed to a philosophical position known as haecceitism, according to which two situations can differ even if they are qualitatively identical. Haecceitism has roots that stretch back at least as far as the medieval period, but there have been few defences of it in the recent philosophical literature. In my article, I argue that recent results from statistical physics not only make haecceitism relevant once more, but also provide a strong argument for it.

A generally very successful method from statistical mechanics called dynamical density functional theory (DDFT; see te Vrugt et al. [2020]) turns out to allow ‘hard’ particles in a one-dimensional channel to pass through one another (Schindler et al. [2019]). This result is unphysical and presents a problem for DDFT.

In response, Wittmann et al. ([2021]) developed an improved version of DDFT known as order-preserving dynamics (OPD). Here, the problem of particles passing through one another is avoided. This is done by using the fact that in statistical mechanics the probability of a certain particle configuration depends on its energy. OPD introduces a fictional particle interaction that implies that configurations with the ‘wrong’ order have an infinitely large energy. This then implies that these configurations have a probability of zero. Equivalently, one can just say that whenever we make a statistical average, we assign a probability of zero to these configurations. Computer simulations show that OPD preserves the ‘correct’ particle order and makes better predictions for the total particle density (an empirically observable quantity) than simpler forms of DDFT.

What does this mean for philosophy? OPD is very helpful for thinking about the problem of haecceitism. Let’s return to our one-dimensional system of two particles, with particle A on the left and B on the right. Assigning a probability of zero to the configurations with B on the left requires these configurations to be different from the ones with A on the left, otherwise they could not have different probabilities. But these configurations are observationally indistinguishable, such that any differences are haecceitistic. Consequently, haecceitism appears to be required to make sense of modern approaches in classical statistical mechanics.

This also makes OPD relevant to a long-standing debate in metaphysics and philosophy of language, namely, the issue of rigid designation and modality de re. This issue is typically explained using an example from Kripke ([1980]): A man points at Richard Nixon and says, ‘This man might have lost the election’. What he means is that there is a possible world in which this particular man, Richard Nixon, lost the election—that is, he is making a modal statement de re about Nixon. Many philosophers would object that this depends on how we designate Nixon. If we take ‘this man’ to mean ‘the guy called Nixon’, then there may of course be a possible world where he lost. But if we take it to mean ‘the winner of the election’, then this modal statement is simply wrong: by definition, the winner of the election cannot have lost the election in any possible world. Consequently, the idea of modality de re makes no sense. Kripke responds to this by insisting that the name ‘Nixon’ is a rigid designator, that is, that it designates the same individual (Nixon) across all possible worlds.

We saw that OPD assigns different probabilities to different possible configurations of a many-particle system. These different possible configurations can be thought of as possible worlds. We specify a possible world with the position and momentum of each particle in it. The question is now whether the label ‘particle A’ is to be understood as a definite description (standing for ‘the particle that is on the left’) or as a rigid designator. If we understand it as a definite description, then we cannot meaningfully speak of configurations in which particle A is on the right. But statistical mechanics requires us to speak about these configurations (and OPD in particular requires us to assign different probabilities depending on whether particle A is on the left or the right). Consequently, particle labels have to be thought of as rigid designators. The successes of statistical mechanics in general and OPD in particular provide a strong argument for modality de re.

One question comes to mind now: OPD is a theory from classical physics that treats particles as ‘hard’ with well-defined positions that move around and bump into one another. As we now know, the world is, at a fundamental level, better understood through quantum mechanics, which describes particles in a very different way, based on wave functions. In quantum mechanics, it is an essential principle that an exchange of two indistinguishable particles cannot make an observable difference. Shouldn’t we then get our metaphysics from quantum rather than classical statistical mechanics if the former is, in some sense, more fundamental?

There are two responses to this. First, it had been thought that quantum mechanics was not the first theory to assume that an exchange of indistinguishable particles can make no observable difference. In his famous article on haecceitism in classical statistical mechanics, Huggett ([1999]) argues that an exchange of indistinguishable particles makes no difference to the observed statistics, and he concludes that classical statistical mechanics is metaphysically neutral regarding haecceitism. What OPD now shows is that particle exchanges do in fact matter in classical physics: if we want to make a case for anti-haecceitism based on physics, we have to do so based on quantum rather than classical mechanics. And this provides an important insight into what is (and is not) novel about the metaphysical implications of quantum mechanics. Second, we can question why our metaphysical inferences should be based solely on microscopic physics. If classical statistical mechanics is a good description for macroscopic objects, then we might also base the metaphysics of macroscopic objects on classical physics. Thus, haecceitism might hold for classical particles in an emergent way.

What does this mean for physics?  From a physicist’s point of view, one could ask whether it is possible in principle to correctly describe a two-particle system in a theory without the kind of asymmetry discussed here. (This, of course, is also of some importance for the force of the philosophical argument.) DDFT predicts that in a system in equilibrium, we are equally likely to find particle A and particle B at a certain position, which is evidently wrong. However, DDFT is an exact theory for systems in equilibrium, so what DDFT says about the equilibrium state cannot be wrong. Thus, from the perspective of a non-haecceitistic version of DDFT, we have no choice but to say that our one-dimensional system never reaches an equilibrium state. This stands in contrast to a very fundamental principle of thermodynamics (which has also led to a considerable debate in philosophy of physics; see te Vrugt [2021]), namely, that every isolated system spontaneously approaches equilibrium (Brown and Uffink [2001]).

OPD has no such problem. Here, the equilibrium state is just the ordered state that the system actually evolves to. This issue can be illustrated using what Schindler et al. ([2019]) call the ‘inverse Gibbs paradox’. We take our one-dimensional, two-particle system and put it in the physical equilibrium state in which A is on the left and B is on the right. This will not be the equilibrium state of a physical theory that does not distinguish between indistinguishable particles (such as ordinary DDFT). Consequently, such a theory is found to predict that the system moves out of the physical equilibrium state into a state where we have the same probability of finding each particle at a given position.

This leaves us with two options. First, we can give up (for these systems) the idea that every isolated system (that has not been prepared in a very specific initial state) goes to equilibrium. Second, we can re-formulate the concept of equilibrium in a haecceitistic way that allows us to recover this principle (this is what is done in OPD). Consequently, whether we allow for haecceitistic differences also influences whether such a one-dimensional system approaches equilibrium.

Listen to the audio essay


te Vrugt, M. [2025]: ‘How to Distinguish between Indistinguishable Particles’, British Journal for the Philosophy of Science, 76, doi: 10.1086/718495

Michael te Vrugt
Westfälische Wilhelms-Universitat Münster


Brown, H. R. and Uffink, J. [2001]: ‘The Origins of Time-Asymmetry in Thermodynamics: The Minus First Law’, Studies in History and Philosophy of Modern Physics, 32, pp. 525–38.

Huggett, N. [1999]: ‘Atomic Metaphysics’, Journal of Philosophy, 96, pp. 5–24.

Kripke, S. A. [1980]: Naming and Necessity, Cambridge, MA: Harvard University Press.

Schindler, T., Wittmann, R. and Brader, J. M. [2019]: ‘Particle-Conserving Dynamics on the Single-Particle Level’, Physical Review E, 99, p. 012605.

te Vrugt, M. [2021]: ‘The Five Problems of Irreversibility’, Studies in History and Philosophy of Science, 87, pp. 136–146.

te Vrugt, M., Löwen, H. and Wittkowski, R. [2020]: ‘Classical Dynamical Density Functional Theory: From Fundamentals to Applications’, Advances in Physics, 69, pp. 121–47.

Wittmann, R., Löwen, H. and Brader, J. M. [2021]: ‘Order-Preserving Dynamics in One Dimension—Single-File Diffusion and Caging from the Perspective of Dynamical Density Functional Theory’, Molecular Physics, 119, p. e1867250.

© The Author (2021)


te Vrugt, M. [2025]: ‘How to Distinguish between Indistinguishable Particles’, British Journal for the Philosophy of Science, 76, doi: 10.1086/718495