Statistical mechanics provides a systematic approach for predicting the behavior of systems when only partial information is present. The key assumption underlying all of statistical mechanics is that every allowed configuration of the system, called a microstate, occurs with equal probability. This approach, valid for both classical and quantum systems, requires only minimal additional information for it to be astonishingly predictive. On page 207 of this issue, Langen *et al.* (*1*) describe a system in which this paradigm is violated and conventional statistical mechanics fails; almost all of the usually allowed microstates are inaccessible via the system's dynamics.

In a classical gas, a microstate describes the position and momentum of every constituent particle. In practical experiments, such complete information is unavailable. Instead, we might wish to know aggregate properties, such as the typical number of particles at some position (the spatial distribution function). The problem is how to predict these properties. Each specific distribution can be described equally well by a great many microstates; in statistical mechanics, we select the distribution that is compatible with as many microstates as possible, thereby maximizing the entropy.

Still, slightly more information is required to make useful predictions. For an isolated system, the total internal energy must be conserved. Of all possible microstates, almost none have any given energy. The allowed microstates are now selected from a reduced set called the microcanonical ensemble. The entropy must be maximized at fixed energy, which is achieved by minimizing the free energy. This constrained minimization is performed by introducing a new a parameter called a Lagrange multiplier (in this case, the temperature). The key point is that while the system's total energy is constrained, the energy of any single particle is not.

This concept is well illustrated by a single particle moving in a one-dimensional (1D) harmonic potential—for example, describing the oscillatory motion of a pendulum. The spatial distribution of this single particle (see the figure, panel A) is strongly peaked, resulting from the constraint that the dynamics of just one particle is nearly fully determined by its energy. The corresponding distributions for 2, 5, and 50 particles, with the same average per-particle energy but with only the total energy constrained, show that these distributions approach the well-known Maxwell-Boltzmann distribution depicted by the pink curve (see the figure, panel B).

Because of their remarkable environmental isolation, ultracold atoms can have additional conserved quantities. Good examples include the conserved angular momentum of rotating atomic gases in highly symmetric potentials, or the conserved magnetization of bosonic and fermionic systems. These constraints have allowed experimental access to new phases of matter (*2*, *3*). Each of the symmetry-induced conservation laws contributes new terms into the suitable free energy. For example, conserved angular momentum introduces an effective magnetic field, whereas conserved magnetization introduces a spin-dependent chemical potential. In these examples, a particular symmetry of the system adds a single new constraint in total, not per particle.

Langen *et al.* studied a 1D atomic Bose gas in which the standard thermodynamic paradigm fails and the internal dynamics contains hidden conserved quantities with the number of constraints proportional to the system size; such a system is said to be integrable. The so-called generalized Gibbs ensemble (GGE) (*4*) is the thermodynamic ensemble that incorporates all of these conserved quantities. Each conserved quantity is associated with its own temperature; a 5000-particle system might require 5000 different temperatures to specify the distribution function.

Our harmonic oscillator example for 5 and 500 particles, where each oscillator separately conserves energy, shows two central points (see the figure, panel C). First, these distributions look nothing like the standard Maxwell-Boltzmann distribution (pink); second, the distributions depend on the system's initial conditions, with peaks set (as in panel A) by the initial energy of each particle. If the per-particle energy had been distributed differently between particles, then the distribution would be peaked at different points. This initial state effect was first observed in a 1D Bose gas (*5*) in which an atomic “Newton's cradle” was created.

The physical origin of these conservation laws is simple and depends on the central assumption that each microstate is occupied with equal probability. Usually, physicists argue that collisions between particles are effective in moving the system among all allowed microstates. For 1D systems with binary, local, and elastic collisions (such as between billiard balls), this assumption fails. If we consider just two particles with momentum *p*_{1} and *p*_{2} colliding at some position *x*, then there are only two possible outcomes that conserve energy and momentum: Either the dynamics continues unchanged (*p*_{1} → *p*_{1} and *p*_{2} → *p*_{2}), or it continues with the roles of the particles swapped (*p*_{1} → *p*_{2} and *p*_{2} → *p*_{1}). In either case, the distribution function is unaltered by the collisions.

Langen *et al.* first prepared individual ultracold 1D Bose gases with about 5000 atoms and split these into two parallel decoupled 1D systems, mapping each atom into a coherent superposition of being in the left and the right system. As a result, each of these 1D subsystems was initialized very far from equilibrium. The quantum mechanical phase difference was then observed along the length of the split system as it approached equilibrium. Langen *et al.* found that after some time the split system reached an equilibrium state, but not one predicted by a standard thermodynamic ensemble.

To identify the details of their experiment's underlying ensemble, the authors made a comparison with the higher-order correlations of the measured phase difference (essentially asking how the phase differences at various points are related) up to 10th order. These distributions were found to be qualitatively different from those predicted by standard thermodynamics. Instead, Langen *et al.* considered the resulting quasi-equilibrium in terms of the predictions of the GGE and found that two separate temperatures were required before the ensemble was consistent with their experimental data within the technical noise.

According to the eigenstate thermalization hypothesis (*4*), many-particle quantum systems generically evolve to distribution functions resembling the microcanonical ensemble for local observables, such as the spatial distribution functions discussed here. It requires special effort, such as in Langen *et al.*'s experiment, to find initial states going beyond this.

Langen *et al.* have shown that cold-atom experiments can be used to study the thermodynamics of systems with internal constraints and can provide the understanding required to predict the outcome of an experiment. One might have expected some 5000 constraints to be required to reconcile experiment and theory; experimentally one was insufficient, but two sufficed. In light of the eigenstate thermalization hypothesis, can the impact of all of these constraints ever be felt, or is a small subset always sufficient to describe the outcome of realistic experiments?

## References and Notes

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**Acknowledgments:**I thank M. Rigol for helpful conceptual discussions. Supported by Army Research Office's atomtronics Multidisciplinary University Research Initiative (MURI), Air Force Office of Scientific Research Quantum Matter MURI, NIST, and the NSF through the Physics Frontier Center at the Joint Quantum Institute.