Exact finite-time response on a compact Hamiltonian phase

The perturbed field is the smooth time-dependent field $X_H-ϵh(t)X_B$ by [F2]. Cover compact $M$ by finitely many smaller coordinate neighborhoods whose closures lie in larger coordinate charts. The coordinate fields and their first derivatives are bounded on these closures, uniformly for $t\in [0,T]$ and $ϵ$ in a fixed compact neighborhood of zero. Local ODE existence therefore gives a common positive continuation time from every point in each smaller neighborhood. Taking the minimum over the finite cover extends every solution across $[0,T]$: a purported finite endpoint can be restarted from a point within that common time of the endpoint. The same argument applies backward for the unperturbed flow. The Euclidean smooth-parameter theorem [F3], applied in these charts on successive common time intervals, gives the smooth parameter variation $\xi (s)={\mathrm{\partial }}_{ϵ}{z}_{ϵ}(s;z){\mathrm{\mid }}_{ϵ=0}\in T_{{\mathrm{\Phi }}_{s}z}M$ with $\xi (0)=0$. In a chart $x$, write $X_H^x$ and $X_B^x$ for the coordinate components. There the ordinary variational equation is ${\dot{\xi }}^a(s)={\mathrm{\partial }}_b{X}_H^{x,a}({\mathrm{\Phi }}_{s}z){\xi }^b(s)-h(s)X_B^{x,a}({\mathrm{\Phi }}_{s}z)\mathrm{.}$ Here ${\mathrm{\partial }}_b{X}_H^{x,a}$ is the matrix of derivatives of coordinate components in that chart; it is not asserted to be a canonical tangent endomorphism. If $y$ is another chart and $\chi =y\circ x^{-1}$ is the transition map, then ${\xi }^y=D\chi {\xi }^x$ and $X_H^y\circ \chi =D\chi X_H^x$, with the analogous identity for $X_B$. Differentiating these identities along the reference path shows that the two variational equations agree: the extra term is $D^2\chi [X_H^x,{\xi }^x]$, which equals $D^2\chi [{\xi }^x,X_H^x]$ by symmetry of the second derivative. Thus the variation is intrinsically defined. The tangent map $Y(t,s)=D{\mathrm{\Phi }}_{t-s}{\mathrm{\mid }}_{{\mathrm{\Phi }}_{s}z}:T_{{\mathrm{\Phi }}_{s}z}M\to T_{{\mathrm{\Phi }}_{t}z}M$ is likewise intrinsic. In chart coordinates, variation of constants for the local linear variational equations, composed across the finite chart intervals, gives the intrinsic identity $\xi (t)=-{\int }_0^{t}h(s)D{\mathrm{\Phi }}_{t-s}{\mathrm{\mid }}_{{\mathrm{\Phi }}_{s}z}{X}_B({\mathrm{\Phi }}_{s}z)ds,$ where the integral is in the fixed vector space $T_{{\mathrm{\Phi }}_{t}z}M$. Applying the differential of $A$ and the chain rule [F8] gives ${{\mathrm{\partial }}_{ϵ}A(z_{ϵ}(t;z))\mid }_0=-{\int }_0^{t}h(s)[X_B(A\circ {\mathrm{\Phi }}_{t-s})]({\mathrm{\Phi }}_{s}z)ds\mathrm{.}$

All integrands in step 1.1 and their parameter difference quotients are bounded on the compact manifold and finite time slab. Thus [F7] passes the derivative through the initial-law integral. Invariance of ${\mu }_{\beta }$ under ${\mathrm{\Phi }}_s$ from [F5]–[F6] then gives ${\frac{d}{dϵ}⟨A(t){⟩}_{ϵ}\mid }_0=-{\int }_0^{t}h(s){\int }_M{X}_B(A\circ {\mathrm{\Phi }}_{t-s})d{\mu }_{\beta }ds\mathrm{.}$

Write ${\rho }_{\beta }=Z_{\beta }^{-1}{e}^{-\beta H}$. For any smooth $G$, let ${\mathrm{\Psi }}_{\tau }$ be the flow of $X_B$. By [F4], ${\mathrm{\Psi }}_{\tau }$ preserves $m_0$. Consequently ${\int }_{M}G({\mathrm{\Psi }}_{\tau }z){\rho }_{\beta }(z)d{m}_0(z)={\int }_{M}G(y){\rho }_{\beta }({\mathrm{\Psi }}_{-\tau }y)d{m}_0(y)\mathrm{.}$ Differentiate at $\tau =0$; compactness bounds the derivatives, so differentiation under the integral is justified by [F7]. This proves $\int {\rho }_{\beta }{X}_{B}Gd{m}_0=-\int G{X}_B{\rho }_{\beta }d{m}_0$. From [F1], $X_{B}H=-X_{H}B$, hence $X_B{\rho }_{\beta }=-\beta {\rho }_{\beta }{X}_{B}H=\beta {\rho }_{\beta }{X}_{H}B$. Therefore $-{\int }_M{X}_{B}Gd{\mu }_{\beta }=\beta {\int }_{M}G(X_{H}B)d{\mu }_{\beta }\mathrm{.}$

Apply step 2.2 with $G=A\circ {\mathrm{\Phi }}_{t-s}$ in step 2.1. This gives $R_{AB}(t-s)$ and proves the stated response formula.

For the correlation, differentiate under the integral on compact $M$: $C_{BA}^{\mathrm{\prime }}(u)=\int (B-⟨B{⟩}_{\beta })X_H(A\circ {\mathrm{\Phi }}_u)d{\mu }_{\beta }$. Applying the same measure-preserving-flow argument as step 2.2 with $X_H$ and using $X_H{\rho }_{\beta }=0$ gives $C_{BA}^{\mathrm{\prime }}(u)=-\int (X_{H}B)(A\circ {\mathrm{\Phi }}_u)d{\mu }_{\beta }$; the constant $⟨A{⟩}_{\beta }$ contributes zero since $\int X_{H}Bd{\mu }_{\beta }=0$. For constant $h_0$, step 3.1 is therefore $-\beta h_0{\int }_0^t{C}_{BA}^{\mathrm{\prime }}(u)du=\beta h_0[C_{BA}(0)-C_{BA}(t)]$. If $C_{BA}(t)\to 0$, the limit is $\beta h_0{C}_{BA}(0)=\beta h_0{\mathrm{Cov}}_{{\mu }_{\beta }}(B,A)$.