Mean field limit of bosonic systems in partially factorized states and their linear combinations

The preprint in pdf is available at arXiv.org.

The bibtex entry for the article can be downloaded here.

We would like to extend the convergence results above to other types of states (that include \(\varphi^{\otimes_n}\) as a particular case). We will use the Fock space method, so we recall some basic concepts of Fock spaces.

Let \(\mathscr{H}\) be a separable Hilbert space, with scalar product

\begin{equation*} \braket{f}{g}_{\mathscr{H}}=\int d{x}\,\bar{f}(x)g(x)\; . \end{equation*}

From \(\mathscr{H}\), construct the symmetric Fock space \(\mathscr{F}_s(\mathscr{H})\) as follows. Define:

\begin{gather*} \mathscr{H}_0=\mathbb{C}\; ,\;\mathscr{H}_n:=S_n\Bigl(\mathscr{H}\otimes\dotsm\otimes\mathscr{H}\Bigr) \end{gather*}

with \(S_n\) orthogonal symmetrizer on the \(n\)-th fold tensor product of \(\mathscr{H}\), i.e.

\begin{gather*} \mathscr{H}_n=\Bigl\{\phi_n(x_1,\dotsc,x_n)\Bigl |\text{ $\phi_n$ is invariant for any permutation of variables}\Bigr\}\; . \end{gather*}

Then

\begin{gather*} \mathscr{F}_s(\mathscr{H}):=\bigoplus_{n=0}^\infty \mathscr{H}_n\; . \end{gather*}

Let \(\phi=(\phi_0,\phi_1,\dotsc,\phi_n,\dotsc)\) be a vector of \(\mathscr{F}_s(\mathscr{H})\). Then \(\mathscr{F}_s(\mathscr{H})\), endowed with the norm

\begin{equation*} \norm{\phi}:=\Bigl(\sum_{n=0}^\infty\norm{\phi_n}^2_{\mathscr{H}_n} \Bigr)^{1/2} \end{equation*}

is a (separable) Hilbert space. The vector \(\Omega=(1,0,0,\dotsc)\) plays a special role in Fock spaces and it is often called the vacuum.

The basic operators of Fock spaces are the annihilation and creation operators. They are the adjoint of one another and are defined as following. Let \(f\in\mathscr{H}\), \(\phi=(\phi_0,\phi_1,\dotsc,\phi_n,\dotsc)\); then

\begin{align*} a(f)&=\int\ide{x}f(x)a(x)\\ a^*(f)&=\int\ide{x}f(x)a^*(x)\; ; \end{align*}

with

\begin{align*} (a(x)\phi)_n(x_1,\dotsc,x_n)&=\sqrt{n+1}\phi_{n+1}(x,x_1,\dotsc,x_n)\\ (a^*(x)\phi)_n(x_1,\dotsc,x_n)&=\frac{1}{\sqrt{n}}\sum_{i=1}^n\delta(x-x_j)\phi_{n-1}(x_1,\dotsc,\hat{x}_j,\dotsc,x_n)\; , \end{align*}

where \(\hat{x}_i\) means such variable is omitted. The \(a^\#\) satisfy the following commutation properties:

\begin{equation*} [a(x),a^*(x')]=\delta(x-x')\; ,\; [a(x),a(x')]=[a^*(x),a^*(x')]=0\; . \end{equation*}

From annihilation and creation operators we can construct an unitary operator, called the Weyl operator that plays a crucial role in our treatment of mean field limits. Let \(\alpha\in\mathscr{H}\); then the Weyl operator, denoted by \(C(\alpha)\), is defined as follows:

\begin{equation*} C(\alpha)=\exp\bigl(a^*(\alpha)-a(\bar{\alpha})\bigr)\; . \end{equation*}

For all \(\alpha\in\mathscr{H}\), \(C(\alpha)\) is unitary on \(\mathscr{F}_s(\mathscr{H})\). In addition to its unitarity, we will use the following properties:

1. \(C^*(\alpha)a(x)C(\alpha)=a(x)+\alpha(x)\), and therefore \(C^*(\alpha)a^*(x)C(\alpha)=a^*(x)+\bar{\alpha}(x)\);
2. \(C(\alpha)C(\beta)=C(\alpha+\beta)e^{-i\Im \braket{\alpha}{\beta}}\);
3. \(C(\alpha)=e^{-\lVert\alpha\rVert^2/2}\exp\{a^*(\alpha)\}\exp\{-a(\alpha)\}\).

Another useful class of operators are the ones defined by the so-called second quantization \citep[see e.g. ReeSim 1975 Chapter X.7]. Let \(A\) be a self-adjoint operator on \(\mathscr{H}\), with domain of essential self-adjointness \(D\). We define the second quantization of \(A\), denoted by \(\de\Gamma(A)\), as the following operator of \(\mathscr{F}_s(\mathscr{H})\): let \(\phi=(\phi_0,\phi_1,\dotsc,\phi_n,\dotsc)\in\mathscr{F}_s(\mathscr{H})\), then

\begin{equation*} (\de\Gamma(A)\phi)_n(x_1,\dotsc,x_n)=\sum_{i=1}^n A(x_i)\phi_n(x_1,\dotsc,x_n)\; , \end{equation*}

where \(A(x_i)\) denotes the operator \(A\) acting on the subspace of \(\mathscr{H}_n\) corresponding to the \(i\)-th variable. \(\de\Gamma(A)\) is essentially self-adjoint on the domain

\begin{equation*} D_A=\Bigl\{ \phi\in\mathscr{F}_s(\mathscr{H})\Bigl | \exists \tilde{n}, \forall n\geq\tilde{n} \; \phi_n=0\; ;\; \forall m\in\mathbb{N} \; \phi_m\in D\otimes\dotsm \otimes D \Bigr\}\; . \end{equation*}

The last notion we introduce is that of number operator \(N\). For all \(n\in\mathbb{N}\), every \(\phi_n\in\mathscr{H}_n\) is an eigenvector of \(N\) with eigenvalue \(n\). Precisely, let \(\phi=(\phi_0,\phi_1,\dotsc,\phi_n,\dotsc)\in\mathscr{F}_s(\mathscr{H})\), then \(N\) is defined as

\begin{equation*} (N\phi_n)_n(x_1,\dotsc,x_n)=n\phi_n(x_1,\dotsc,x_n)\; . \end{equation*}

\(N\) is a self-adjoint operator with domain

\begin{equation*} D(N)=\Bigl\{\phi\in\mathscr{F}_s(\mathscr{H}),\sum_{n=0}^\infty n^2\norm{\phi_n}^2<\infty\Bigr\}\; ; \end{equation*}

it satisfies the following properties:

1. \(N=\de\Gamma(1)=\int\ide{x}a^*(x)a(x)\).

2. Let \(f\in\mathscr{H}\); then for all \(\phi\in D(N^{1/2})\):

   \begin{equation*} \norm{a^\#(f)\phi}\leq \norm{f}_{\mathscr{H}}\norm{(N+1)^{1/2}\phi}\; . \end{equation*}

Furthermore, let \(E_N(\lambda)\) be the spectral family of the operator \(N\). Then for all \(E\)-measurable operator valued function \(g\), we have

\begin{gather*} g(N)a(x)=a(x)g(N-1)\\ g(N)a^*(x)=a^*(x)g(N+1)\; ; \end{gather*}

on suitable domains.

1. Let \(f\in\mathscr{H}\otimes\mathscr{H}\); then for all \(\phi\in D(N)\):

   \begin{align*} \norm{\int\de{x}\ide{y}f(x,y)a^\#(x)a^\#(y)\phi}&\leq \norm{f}_{\mathscr{H}\otimes\mathscr{H}}\norm{(N+1)\phi}\\ \norm{\int\de{x}\ide{y}f(x,y)a^*(x)a(y)\phi}&\leq \norm{f}_{\mathscr{H}\otimes\mathscr{H}}\norm{N\phi}\; . \end{align*}

The quantum system we would like to study describes \(n\) non-relativistic interacting bosons; its dynamics is dictated by the following Hamiltonian of \(L^2_s(\mathbb{R}^{3n})\): let \(V\) be a real and symmetric function of \(\mathbb{R}^3\) (other assumptions on the potential will be specified later), then

\begin{equation} \label{eq:10} H_n=H_{0n}+V_n=\sum_{j=1}^n-\Delta_{x_j}+\frac{1}{n}\sum_{i< j}^n V(x_i-x_j)\; . \end{equation}

This operator can be written in the language of second quantization on \(\mathscr{F}_s(L^2(\mathbb{R}^3))\) as:

\begin{equation} \label{eq:2} H=\int\ide{x}(\nabla a)^*(x)\nabla a(x)+\frac{1}{2n}\int\de{x}\ide{y} V(x-y)a^*(x)a^*(y)a(x)a(y)\; ; \end{equation}

\(H_n\) and \(H\) agree on \(\mathscr{H}_n\). Let now \(\phi\in\mathscr{H}_n\); we denote by \(\phi(t)\) the time evolution of \(\phi\) and by

\begin{equation*} \rho_\phi(t)=\ketbra{\phi(t)}{\phi(t)} \end{equation*}

the corresponding density matrix, with integral kernel

\begin{equation*} \rho_\phi(t,x_1,\dotsc,x_n;y_1,\dotsc,y_n)=\phi(t,x_1,\dotsc,x_n)\bar{\phi}(t,y_1,\dotsc,y_n)\; . \end{equation*}

Also, we denote by \(\Tr_1\rho_{\phi}(t)\) the one-particle reduced density matrix, with integral kernel

\begin{equation*} \Tr_1\rho_{\phi}(t,x;y)=\int\de{x_1}\dotsm\ide{x_{n-1}}\phi(t,x,x_1,\dotsc,x_{n-1})\bar{\phi}(t,y,x_1,\dotsc,x_{n-1})\; . \end{equation*}

In the mean field limit \(n\to\infty\), using suitable initial states, the one-particle reduced density matrix is expected to converge in some sense to the solution of Hartree equation:

\begin{equation} \label{eq:4} i\partial_t \varphi_t = -\Delta\varphi_t +(V*\ass{\varphi_t}^2)\varphi_t\; . \end{equation}

As discussed above, such convergence has already been proved for factorized (\(\varphi^{\otimes_n}\)) and coherent (\(C(\sqrt{n}\varphi)\Omega\)) states. In this paper we prove that mean field limit convergence can be obtained for a wider class of states. First of all we consider states with \(n\) particles, only a fraction of which is factorized in the same state \(\varphi\) [see also LewNamSerSol 2013 where these states are used to construct an isomorphism between \(\mathscr{H}_n\) and the truncated Fock space orthogonal to \(\varphi\)]. The precise definition is the following:

• Definition 1. [\(\theta_{n,m}\)]. Let \(\mathscr{H}\) be a Hilbert space; \(\mathscr{F}_s(\mathscr{H})=\bigoplus_{n=0}^\infty\mathscr{H}_n\) the corresponding symmetric Fock space. We also denote by \(\mathscr{D}\subseteq\mathscr{H}\) a subspace of \(\mathscr{H}\).

  Now let \(\varphi\in \mathscr{D}\) such that \(\braket{\varphi}{\varphi}_{\mathscr{H}}=1\) and \(\psi_m\in\mathscr{H}_m\) such that \(\norm{\psi_m}_{\mathscr{H}_m}=1\) and

  \begin{equation} \label{eq:1} \braket{\varphi}{\psi_m}_{\mathscr{H}_1}=\int\ide{x}\bar{\varphi}(x)\psi_m(x,x_1,\dotsc,x_{m-1})=0\; . \end{equation}

  We define

  \begin{equation*} \theta_{n,m}:= c_{n,m}S_{n}(\varphi^{\otimes_{n-m}}\otimes\psi_m)\in\mathscr{H}_n\; ; \end{equation*}

  where:

  \begin{gather*} \varphi^{\otimes_j}=\underset{j}{\underbrace{\varphi\otimes\dotsm\otimes\varphi}}\; ,\\ S_j:\underset{j}{\underbrace{\mathscr{H}\otimes\dotsm\otimes\mathscr{H}}}\longrightarrow \mathscr{H}_j\text{ orthogonal projector (symmetrizer),}\\ c_{n,m}=\binom{n}{m}^{1/2}\text{ (such that $\norm{\theta_{n,m}}=1$).} \end{gather*}

The convergence result about $θ$-vectors is formulated in the following theorem:

• Theorem 1. Suppose there exists \(D>0\) such that the operator inequality

  \begin{equation*} V^2(x)\leq D(1-\Delta_x) \end{equation*}

  is satisfied. Let \(\theta_{n,m}\) satisfy definition 1 with \(\varphi\in H^1(\mathbb{R}^3)\). Also, let \(\varphi_t\) be the \(\mathscr{C}^0(\mathbb{R},H^1(\mathbb{R}^3))\) solution of Hartree equation with initial datum \(\varphi(0)=\varphi\). Then \(\forall t \in\mathbb{R}\):

  \begin{equation*} \Tr \ass{\Tr_1\rho_{\theta_{n,m}}(t)-\ketbra{\varphi_t}{\varphi_t}}\leq 2 K_1 e^{K_2\ass{t}}\frac{1}{\sqrt{n}}e^{m/2}(m+1)^7\; ; \end{equation*}

  where the \(K_i\), \(i=1,2\), are positive and depend only on \(D\) and \(\norm{\varphi}_{H^1}\).

• Remark. For technical reasons (see lemma 2 below), the result above holds only if \(m\leq \sqrt{7+3n}-3\). However since we are considering the limit \(n\to\infty\), a stricter bound for \(m\) has to be imposed if we want convergence to hold. In particular,

  \begin{equation*} \underset{n\to\infty}{\text{Tr$-$lim}}\; \Tr_1\rho_{\theta_{n,m}}(t) = \ketbra{\varphi_t}{\varphi_t} \end{equation*}

  whenever exists \(0\leq a<1\) such that, for large \(n\),

  \begin{equation*} m\sim a\ln n\; . \end{equation*}

Also, we study the mean field limit for superpositions of \(\varphi^{\otimes_n}\), \(\theta_{n,m}\) or \(C(\sqrt{n}\varphi)\Omega\) states. Such linear combinations have to satisfy the following definition:

• Definition 2. [\(\Phi\), \(\Theta\), \(\Psi\)] Let \(\mathscr{H}\) be a Hilbert space; \(\mathscr{F}_s(\mathscr{H})=\bigoplus_{n=0}^\infty\mathscr{H}_n\) the corresponding symmetric Fock space. We also denote by \(\mathscr{D}\subseteq\mathscr{H}\) a subspace of \(\mathscr{H}\). Furthermore, assume:

  1. \((\alpha_i)_{i\in \mathbb{N}},(\beta_i)_{i\in \mathbb{N}},(\gamma_i)_{i\in \mathbb{N}}\in l^1\).
  2. \((\varphi^{(i)})_{i\in\mathbb{N}}\) such that \(\varphi^{(i)}\in \mathscr{D}\), \(\forall i\in\mathbb{N}\) and \(\sup_{i\in\mathbb{N}}\norm{\varphi^{(i)}}_{\mathscr{D}}= M<+\infty\). Also we ask either
     1. \(\norm{\varphi^{(i)}}_{\mathscr{H}}=1\), \(\varphi^{(i)}\) and \(\varphi^{(j)}\) linearly independent for all \(i\neq j\); or
     2. \(\varphi^{(i)}\neq\varphi^{(j)}\) for all \(i\neq j\).
  3. To each vector of \((\varphi^{(i)})_{i\in\mathbb{N}}\) satisfying 2.1) we associate \((\varphi^{(i)})^{\otimes_n}\in\mathscr{H}_n\) and \(\theta^{(i)}_{n,m_i}\) satisfying definition 1, such that \(m_i\leq m_j\) if \(i\leq j\), \(m_i\leq m\) for all \(i\in\mathbb{N}\). To each vector of \((\varphi^{(i)})_{i\in\mathbb{N}}\) satisfying 2.2) we associate the Weyl operator \(C(\sqrt{n}\varphi^{(i)})\).

  Then define

  \begin{equation*} \Phi=\sum_{i\in\mathbb{N}}\alpha_i(n)\, (\varphi^{(i)})^{\otimes_n}\; ,\; \Theta=\sum_{i\in\mathbb{N}}\beta_i(n)\, \theta_{n,m_i}^{(i)}\; \Psi=\sum_{i\in\mathbb{N}}\gamma_i(n) C(\sqrt{n}\varphi^{(i)})\Omega\; . \end{equation*}

  The suites \((\alpha_i(n))_{i\in\mathbb{N}}\), \((\beta_i(n))_{i\in\mathbb{N}}\) and \((\gamma_i(n))_{i\in\mathbb{N}}\) are chosen such that \(\norm{\Phi}=\norm{\Theta}=\norm{\Psi}=1\).

• Remark 1. In particular we have

  \begin{align*} \alpha_i(n)&=\alpha_i\Bigl(\sum_{i,j\in\mathbb{N}}\bar{\alpha}_i\alpha_j \braket{\varphi^{(i)}}{\varphi^{(j)}}^n\Bigr)^{-1/2}\; ,\\ \beta_i(n)&=\beta_i\Bigl(\sum_{i,j\in\mathbb{N}}\bar{\beta}_i\beta_j \braket{\theta_{n,m_i}^{(i)}}{\theta_{n,m_j}^{(j)}}\Bigr)^{-1/2}\; ,\\ \gamma_i(n)&=\gamma_i\Bigl(\sum_{i,j\in\mathbb{N}}\bar{\gamma}_i\gamma_j \braket{C(\sqrt{n}\varphi^{(i)})\Omega}{C(\sqrt{n}\varphi^{(j)})\Omega}\Bigr)^{-1/2}\; . \end{align*}

  For all \(i\in\mathbb{N}\), \(\alpha_i(n)\), \(\beta_i(n)\) and \(\gamma_i(n)\) are convergent when \(n\to\infty\), and their absolute value is uniformly bounded in \(n\) respectively by \(K_\alpha\ass{\alpha_i}\), \(K_\beta\ass{\beta_i}\) and \(K_\gamma\ass{\gamma_i}\), where the constants depend only on the $l¹$-norm of the respective suite. Further details are discussed in section IV.B.

• Remark 2 Since \(\Psi\) states do not belong to a fixed particle subspace, we define the integral kernel of the reduced density matrix to be

  \begin{equation*} \Tr_1 \rho_{C\Omega}(t,x;y)=\frac{1}{\braket{\Psi(t)}{N\Psi(t)}}\braket{\Psi(t)}{a^*(y)a(x)\Psi(t)}\; . \end{equation*}

With a linear superposition of states as initial condition, the mean field limit is not a pure state. The reduced density matrix converges to a linear combination of projections on the solution of Hartree equation corresponding to each initial datum \(\varphi^{(i)}\); in the topology induced by the Hilbert-Schmidt norm (\(\norm{\,\cdot\,}_{HS}\)):

• Theorem 2 Suppose there exists \(D>0\) such that the operator inequality

  \begin{equation*} V^2(x)\leq D(1-\Delta_x) \end{equation*}

  is satisfied. Let \(\Phi\), \(\Theta\) and \(\Psi\) satisfy definition 2 with \(\mathscr{D}=H^1(\mathbb{R}^3)\). Also, for all \(i\in\mathbb{N}\), let \(\varphi^{(i)}_t\) be the \(\mathscr{C}^0(\mathbb{R},H^1(\mathbb{R}^3))\) solution of Hartree equation with initial datum \(\varphi^{(i)}(0)=\varphi^{(i)}\). Then \(\forall t \in\mathbb{R}\), if there is \(0\leq a<1/2\) such that, for large \(n\), \(m\sim a\ln n\):

  \begin{align*} \underset{n\to\infty}{\text{HS-}\mathrm{lim}}\; &\Tr_1\rho_{\Phi}(t)=\sum_{i\in\mathbb{N}}\ass{\frac{\alpha_i}{\norm{(\alpha_i)_{i\in\mathbb{N}}}_{l^2}}}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}\; ;\\ \underset{n\to\infty}{\text{HS-}\mathrm{lim}}\; &\Tr_1\rho_{\Theta}(t)=\sum_{i\in\mathbb{N}}\ass{\frac{\beta_i}{\norm{(\beta_i)_{i\in\mathbb{N}}}_{l^2}}}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}\; ;\\ \underset{n\to\infty}{\text{HS-}\mathrm{lim}}\; &\Tr_1\rho_{C\Omega}(t)=\sum_{i\in\mathbb{N}}\ass{\frac{\gamma_i}{\norm{(\gamma_i)_{i\in\mathbb{N}}}_{l^2}}}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}\; . \end{align*}

  In particular, the following bounds hold:

  \begin{align} \label{eq:5}\tag{1} &\norm{\Tr_1\rho_{\Phi}(t)-\norm{(\alpha_i)_{i\in\mathbb{N}}}^{-2}_{l^2}\sum_{i\in\mathbb{N}} \ass{\alpha_i}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}}_{HS}\leq K_1\sum_{i < j}\ass{\bar{\alpha}_i(n)\alpha_j(n)} \ass{\braket{\varphi^{(i)}}{\varphi^{(j)}}}^n + K_2 e^{K_3\ass{t}}\frac{1}{n^{1/4}}\; , \\ \label{eq:6}\tag{2} &\norm{\Tr_1\rho_{\Theta}(t)-\norm{(\beta_i)_{i\in\mathbb{N}}}^{-2}_{l^2} \sum_{i\in\mathbb{N}}\ass{\beta_i}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}}_{HS}\leq K_1\sum_{i < j}\ass{\bar{\beta}_i(n)\beta_j(n)}\ass{\braket{\theta_{n,m_i}^{(i)}}{\theta_{n,m_j}^{(j)}}} + K_2 e^{K_3\ass{t}}\frac{e^{m/2}(m+1)^3}{n^{1/4}}\; ; \\ \label{eq:7}\tag{3} &\norm{\Tr_1\rho_{C\Omega}(t)-\norm{(\gamma_i)_{i\in\mathbb{N}}}^{-2}_{l^2} \sum_{i\in\mathbb{N}}\ass{\gamma_i}^2\ketbra{\varphi_t^{(i)}}{\varphi_t^{(i)}}}_{HS}\leq K_1e^{-4M^2 n} + K_2 e^{K_3\ass{t}}\frac{1}{n^{1/2}}\; ; \end{align}

  where the \(K_i\), \(i=1,2,3\), are positive and depend on \(D\), \(M\) and \(\norm{(\,\cdot\,_i)_{i\in\mathbb{N}}}_{l^1}\).

• Remark 3 The first term on the right hand side of equation \eqref{eq:5} converges to zero when \(n\to\infty\) because, by definition 2 and Riesz's lemma, \(\ass{\braket{\varphi^{(i)}}{\varphi^{(j)}}}<1\) (and \(\ass{\bar{\alpha}_i(n)\alpha_j(n)}\leq K_\alpha^2 \ass{\bar{\alpha}_i\alpha_j}\)). The first term on the right hand side of equation \eqref{eq:6} also converges to zero if \(m\leq \ln n\). This is because

  \begin{equation*} \ass{\braket{\theta_{n,m_i}^{(i)}}{\theta_{n,m_j}^{(j)}}}\leq (m+1)(m!)^2 n^m\ass{\braket{\varphi^{(i)}}{\varphi^{(j)}}}^{n-2m}\leq K n^{2(\ln n+1)} \ass{\braket{\varphi^{(i)}}{\varphi^{(j)}}}^{n-2\ln n}\underset{n\to\infty}{\longrightarrow}0\; ; \end{equation*}

  furthermore since \(\ass{\bar{\beta}_i(n)\beta_j(n)}\ass{\braket{\theta_{n,m_i}^{(i)}}{\theta_{n,m_j}^{(j)}}}\leq K_\beta^2\ass{\bar{\beta}_i\beta_j}\) we can exchange summation with the limit \(n\to\infty\).

• Remark 4 In this paper we focused attention only on one-particle reduced density matrices. The same method can be used to calculate the limit of $k$-particle reduced density matrices (with \(k > 1\)).

The rest of the paper is organized as follows. In section II we analyze the dynamics of classical (Hartree) and quantum system. In section III the combinatorial properties of \(\theta_{n,m}\) states are studied. Finally in section IV we consider the limit \(n\to\infty\), and prove theorems 1 and 2.