Mean-Field Limits

A mean-field limit makes a nonlinear one-particle equation emerge from a sequence of linear many-particle Schrödinger evolutions. For bounded pair interactions, factorization of every fixed reduced density matrix propagates for bounded time intervals. This page proves that statement through the reduced-density hierarchy. The mean-field theory article supplies variational interpretation and physical applications; the present result specifies an actual limiting process.

Required background. Many-body Hilbert spaces supplies independent-particle generator domains. Partial trace supplies reduced density matrices and trace duality.

The weak-pair-coupling Hamiltonian

Set $\hbar=1$. Let $h$ be self-adjoint on a separable one-particle Hilbert space $\mathfrak h$, and let $V$ be a bounded self-adjoint operator on $\mathfrak h\otimes\mathfrak h$, invariant under exchanging its two factors. On the symmetric $N$-particle space, for $N\geq2$, define

$$H_N=\sum_{j=1}^N h_j+ \frac1{N-1}\sum_{1\leq i<j\leq N}V_{ij}.$$

The independent-particle sum has its joint-spectral domain. For each fixed $N$, the interaction is bounded, so $H_N$ is self-adjoint on that same domain. The interaction norm can grow linearly in $N$; it need not be uniformly bounded as a many-body operator.

Let $\Gamma_N(t)=e^{-itH_N}\Gamma_N(0)e^{itH_N}$, where $\Gamma_N(0)$ is a positive trace-one bosonic density matrix. Write $\gamma_N^{(k)}$ for its trace-one $k$-particle marginal.

For a unit vector $\varphi$, define the bounded self-adjoint one-particle operator

$$V_\varphi=(I\otimes\langle\varphi|) V(I\otimes|\varphi\rangle).$$

The Hartree evolution is

$$i\partial_t\varphi_t=h\varphi_t+V_{\varphi_t}\varphi_t.$$

For arbitrary unit initial data this is understood in its mild form with the group $e^{-ith}$. The nonlinear term is locally Lipschitz on $\mathfrak h$, and the equation preserves norm. These facts give a unique global mild solution. A differential equation in $D(h)$ requires the corresponding additional regularity.

The convergence statement is about fixed marginals

Assume that for every fixed $k$,

$$\left\|\gamma_N^{(k)}(0)- |\varphi_0^{\otimes k}\rangle \langle\varphi_0^{\otimes k}|\right\|_1 \longrightarrow0.$$

Then, for every fixed $k$ and $T<\infty$,

$$\sup_{|t|\leq T} \left\|\gamma_N^{(k)}(t)- |\varphi_t^{\otimes k}\rangle \langle\varphi_t^{\otimes k}|\right\|_1 \longrightarrow0.$$

Consequently every bounded observable involving a fixed number of particles has the Hartree limiting expectation. The statement does not assert convergence of the entire $N$-particle wave function in norm, uniformity for $k$ proportional to $N$, or uniformity for arbitrarily long times.

The finite hierarchy exposes the error terms

Taking partial traces of the von Neumann equation gives

$$i\partial_t\gamma_N^{(k)} ={}\left[\sum_{j=1}^kh_j+ \frac1{N-1}\sum_{i<j\leq k}V_{ij}, \gamma_N^{(k)}\right] +\frac{N-k}{N-1} \sum_{j=1}^k \operatorname{Tr}_{k+1} [V_{j,k+1},\gamma_N^{(k+1)}].$$

For unbounded $h$, use the integrated equation in the free interaction picture. Its bounded interaction terms act continuously on trace class, so no differentiation of an arbitrary density matrix is required.

The factorized Hartree marginals $\gamma^{(k)}(t)$ satisfy the infinite hierarchy obtained by deleting the internal $1/(N-1)$ interaction and replacing the exterior coefficient by one. This follows by differentiating tensor products for regular data and then using continuity for mild data.

Put $D_k(t)=\|\gamma_N^{(k)}(t)-\gamma^{(k)}(t)\|_1$ and $v=\|V\|$. Unitary invariance, trace-norm contraction of partial trace, and $\|[V,X]\|_1\leq2v\|X\|_1$ give, for $t\geq0$ and $k<N$,

$$D_k(t)\leq D_k(0) +\frac{3v\,k(k-1)t}{N-1} +2vk\int_0^tD_{k+1}(s)\,ds.$$

The finite-size term combines the internal pairs and the difference in the exterior coefficient. This estimate identifies exactly why a hierarchy closed by hand becomes correct only in the limit.

Iteration proves propagation of factorization

Iterate the last integral inequality $L$ times, taking $N>k+L$. For fixed $k,L$ and bounded $t$, all initial-error contributions vanish as $N\to\infty$, and every finite-size contribution is $O(N^{-1})$. The remaining term is bounded using $D_{k+L}\leq2$:

$$2(2vt)^L\frac{k(k+1)\cdots(k+L-1)}{L!} =2(2vt)^L\binom{k+L-1}{L}.$$

For fixed $k$, the binomial factor grows polynomially in $L$. Hence the remainder tends to zero whenever $2vt<1$. Taking $N\to\infty$ first and then $L\to\infty$ proves uniform convergence on a sufficiently short time interval.

At its endpoint all fixed marginals again converge to the Hartree hierarchy. Restarting the same argument covers any prescribed finite interval in finitely many steps. Negative times are treated identically. This completes the proof; no compactness extraction or uniqueness assumption about a formal infinite hierarchy was left implicit.

Boundedness of $V$ is doing real work in the commutator estimate. Coulomb interactions, contact limits, and Gross–Pitaevskii scalings need additional estimates and cannot be inferred by substituting a singular potential into this proof.

A solvable spin model shows the time restriction

Take $\mathfrak h=\mathbb C^2$, $h=0$, $V=gZ\otimes Z$, and initial state $|+\rangle^{\otimes N}$, where $Z|\uparrow\rangle=|\uparrow\rangle$ and $|+\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt2$. Hartree gives zero magnetization and a constant one-particle state. The exact reduced off-diagonal entry is

$$\langle\uparrow|\gamma_N^{(1)}(t)|\downarrow\rangle =\frac12\left(\cos\frac{2gt}{N-1}\right)^{N-1}.$$

It tends to $1/2$ at every fixed time. At times $t=c\sqrt{N-1}$, however, it tends to $\tfrac12e^{-2g^2c^2}$. The finite-time theorem does not control that growing time scale.

Exercises

Derive the spin coherence.

Solution

All $Z_iZ_j$ terms commute. Flipping the first spin changes the phase contributed by each other spin by $\pm2gt/(N-1)$. Tracing each independent initial $|+\rangle$ spin averages those phases to a cosine. The $N-1$ factors multiply. Interactions among the traced spins cancel between bra and ket.

What error controls a bounded measurement? If $\|A_k\|\leq1$, bound its expectation error.

Solution

Trace duality gives $|\operatorname{Tr}A_k(\gamma_N^{(k)}-\gamma^{(k)})|\leq D_k$. Thus the theorem controls every bounded fixed-$k$ observable simultaneously in operator norm. It does not control an unbounded energy observable without a separate uniform-integrability or energy estimate.

References

• F. Golse, “The mean-field limit for the dynamics of large particle systems,” Journées Équations aux Dérivées Partielles, article 9, 1–47, 2003, doi:10.5802/jedp.623.
• H. Spohn, “Kinetic equations from Hamiltonian dynamics: Markovian limits,” Reviews of Modern Physics 52, 569–615, 1980, doi:10.1103/RevModPhys.52.569.