Skip to content

Function icon Equations of motion (current model) and observables

Audience: readers who want the model equations and what is measured/exported.

We evolve correlation \(C(t,t')\) and response \(R(t,t')\) after a quench on the non-equidistant grid \(t\ge t'\). Currently, DYNAMITE has the mixed spherical \(p\)-spin equations hardcoded, matching the definitions in Lang–Sachdev–Diehl (Phys. Rev. Lett. 135, 247101 (2025), doi:10.1103/z64g-nqs6).

Mixed spherical p-spin EOMs (paper definitions)

Let the memory kernels be functionals of C and R with p- and q-body terms and spherical constraint enforced via a Lagrange multiplier µ(t). The equations read schematically

\[ \begin{aligned} \partial_t C(t,t') &= -\mu(t)\, C(t,t') \\ &\quad+ \int_0^t ds\, \Sigma(t,s)\, C(s,t') \\ &\quad+ \int_0^{t'} ds\, D(t,s)\, R(t',s), \\ \partial_t R(t,t') &= -\mu(t)\, R(t,t') + \delta(t-t') \\ &\quad+ \int_{t'}^t ds\, \Sigma(t,s)\, R(s,t'), \end{aligned} \]

with model-specific kernels (for the mixed spherical \(p\)-spin model) of the form

\[ \begin{aligned} \Sigma(t,s) &= f''(C(t,s))R(t,s),\\ D(t,s) &= f'(C(t,s)) + \beta\, \delta(s)\, f'(C(t,0))\,, \end{aligned} \]

where

\[ \begin{aligned} f(x)=J_p^2\, x^p + J_q^2\, x^q\,, \end{aligned} \]

and the Lagrange multiplier of the spherical constraint

\[ \begin{aligned} \mu(t)=T_f+\int_0^t ds\left[\Sigma(t,s)C(t,s)+D(t,s)R(t,s)\right]\,. \end{aligned} \]

Notes

  • The spherical constraint fixes \(\mu(t)\) such that \(C(t,t)=1\).
  • The concrete prefactors and any thermal/noise terms follow the conventions published in Phys. Rev. Lett.; DYNAMITE implements those definitions directly.
  • The exact expressions and units match the paper; see source under include/EOMs/ for the hardcoded operators used at runtime.
  • The non-stationary (aging) regime requires both time integrals and thus benefits from the sparse 2D grid and renormalized history.

Stored fields

  • QKv, QRv: discretized correlation/response on the sparse grid
  • dQKv, dQRv: time derivatives
  • t1grid: time grid values used by the integrator
  • rvec, drvec: reduced observables stored along the diagonal

See include/EOMs/ and include/interpolation/ for algorithmic details.