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In theoretical studies of Bose-Einstein condensates, it's common to look at the one-body density matrix: $$ n_{ab}=\langle\hat{c}_a^\dagger\hat{c}_b\rangle $$ where the $\hat{c}_j$ are annihilation operators for some single-particle modes. (Often, it's written in the position basis as $n(\mathbf{x},\mathbf{y})=\langle\hat{\psi}^\dagger(\mathbf{x})\hat{\psi}(\mathbf{y})\rangle$, but I'm working in a discrete basis.)

The von Neumann entropy of a system is defined as: $$S=-\text{Tr}\left\{\hat{\rho}\log\left(\hat{\rho}\right)\right\}$$ where $\hat{\rho}$ is the full many-body density matrix. I'm working with a method in which I can't tractably obtain the full many-body density matrix, but frequently use the one-body density matrix. Consider the quantity: $$S^{(n)}=-\text{Tr}\left\{\mathbf{n}\log\left(\mathbf{n}\right)\right\}$$ where $\mathbf{n}$ is the one-body density matrix, with matrix elements $n_{ab}$ as defined in my first equation. In the case of a single particle, the one-body density matrix is the full density matrix, and this would give us the von Neumann entropy. What about for a many-body system? Is it meaningful in any way then? Intuition makes me suspect that it's something like a lowest-order approximation to the full von Neumann entropy that discards many-body effects (and my needs make me hope so!), but I'm not sure how I'd go about formally showing this, or deducing its meaning.

user502382
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2 Answers2

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The one-body entropy of a system at equilibrium is equivalent to the thermodynamic entropy per particle. It is ubiquitous in discussions of cooling ultracold gases to reach low-entropy ordered states (here is one example of many), and is often referred to as the "entropy per particle" ($S/N$) or just the "entropy" of the gas. This equivalence is a consequence of the Eigenstate Thermalization Hypothesis.

This is often applied to categorize phases of matter by the entropy required. For example, in a homogeneous Bose gas condensation occurs when $S/N=1.3 k_B$. Granted, this is calculated for a noninteracting gas, but it remains a good approximation for a gas with weak repulsive interactions in which the state no longer factorizes.

Edit: Since the bounty asks for sources, here is an experimental work that explores the correspondence between single-body Von Neumann entropy and thermodynamic entropy at equilibrium. They look at the entropy of a single site, rather than a single particle, but since they consider a uniformly filled system these are interchangeable as $S/V=nS/N$. More generally, the Eigenstate Thermalization Hypothesis that is studied applies to all few-body observables. I have discussed this paper more in a related previous question.

Rococo
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For a system of identical particles, this paper https://journals.aps.org/pre/abstract/10.1103/PhysRevE.81.021119 defines the quantum entropy (Eq. (6)) in terms of the one-body density matrix.

Chaos
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