An interactive guide to RPMD-GLE
From beads
to bath memory.
One particle. Several ways for its ring polymer to move. A different memory kernel for each mode. Follow the steps to see where the A matrix comes from.
How can one ring polymer move?
A quantum particle is represented by P replicas, called beads, joined by harmonic springs. These are samples along an imaginary-time path, not P different atoms. A normal mode is a coordinated displacement of those same beads.
The selected displacement pattern
Beads are arranged on a circle to show their connectivity; every displacement is horizontal. The centroid mode translates the whole ring rigidly. Internal modes move beads by different amounts while keeping the centroid (cross) fixed. The dashed circle shows the undisplaced layout. Amplitude and animation speed are illustrative, not a simulated trajectory.
Free ring-polymer frequencies
These frequencies belong to the connecting springs. They are not the vibrational frequencies of a molecule or surface.
The frequency formula, and what it does not tell you
The centroid is the mean bead position. Mode 0 moves every bead together, so it does not stretch a spring: ω₀ = 0. The physical potential can still exert a force on it. Modes n and P − n have equal frequencies and form two independent real displacement patterns. Here we show a cosine pattern for the first member and a sine pattern for the second.
The free spring frequencies do not depend on atomic mass when the physical masses are used for RPMD. Mass and the physical potential do affect the sampled bead distribution. At a fixed mode index and sufficiently large P, the frequency approaches 2πn kBT/ℏ; increasing P adds higher-frequency modes.
What does the environment remember?
Friction can depend on past motion. The memory kernel K(t) tells us how strongly an earlier velocity affects the force now. Its frequency representation is the friction spectrum Λ(ω). Select a bath, then compare the original spectrum with the spectrum for your selected ring-polymer mode.
One bath, mode-dependent spectra
The same information in time
An ideal delta function has no finite height. It is described here rather than drawn as a finite peak.
How the mode transformation works
Λn(Ω) = [√(Ω² − ωn²) / Ω] Λ(√(Ω² − ωn²))
For the centroid, use Λ₀(Ω) = Λ(Ω), including its value at zero. This transformation shifts each bath frequency ω to √(ω² + ωn²) and changes its weight. An internal mode therefore has a spectral gap below its spring frequency. The original bath is unchanged; its effective action on the ring polymer depends on the mode.
The finite-memory example uses Λ(ω) = η exp(−ω/ωc). Its centroid kernel is K₀(t) = 2ηωc / [π(1 + ωc²t²)]. All kernels retain their amplitudes; none is divided by K(0). Negative lobes describe oscillatory memory and do not imply a negative friction spectrum.
“Ohmic” means J(ω) is proportional to ω at low frequency. An Ohmic bath with a finite cutoff still has memory. An exactly flat Λ over all frequencies is the ideal Markovian limit.
There is also a mean-force correction
Eliminating the bath changes the equilibrium ring-polymer potential as well as its dynamics. In this lesson’s linear, mass-scaled coupling example, the extra potential per mode is ½αnQn².
This is separate from fitting the dynamical kernel. For a nonlinear coupling, the correction involves the normal modes of the coupling function F(q); modes need not evolve independently. The visualizer starts with constant friction / linear coupling to keep that distinction visible.
How can a matrix store a memory?
Introduce a small set of auxiliary variables s. They receive information from the momentum and feed it back later. The full state (p, s) evolves using its present values alone. If we eliminate s, the equation for p contains a memory integral.
One momentum, one auxiliary
dX/dt = −AX + noise, X = (p, s₁, …)
Entries are in fs⁻¹. This is the friction step for mass-scaled variables with constant linear coupling. Conservative forces are separate. Thermal noise is required; the matrix alone is not the complete stochastic model.
Where does the noise go?
In this convention, equilibrium covariance is C = kBT I and BBᵀ = AC + CAᵀ. For the examples here, noise acts on the auxiliaries with amplitude √(2γkBT). The opposite signs of the p–s coupling transfer information without themselves dissipating it. In an RPMD implementation, use the covariance and temperature factor appropriate to its normal-mode normalization.
A matrix produces a spectrum and a kernel
This is a separate sandbox, not a fit to section 2. Change θ to change the amplitude, γ to change the decay, and Ωaux to introduce oscillations.
“Markovian embedding” describes the enlarged state (p, s). The momentum alone can still have non-Markovian dynamics. There is no contradiction: s carries its memory.
How a larger A matrix becomes a kernel
Each damped-cosine term uses a pair of auxiliary variables. Multiple terms share the same momentum and their kernels add. A model with two simple decay terms and four damped-cosine terms has 2 + 2×4 = 10 auxiliaries: its full A matrix is 11 × 11, regardless of how many beads you use.
A finite collection of damped terms approximates a target over a chosen frequency/time range. In particular, it generally leaks a little spectral weight into an internal mode’s exact gap. A good-looking curve is not by itself a production-quality fit.
What does George’s fitter actually fit?
It adjusts auxiliary parameters so that the spectrum generated by A matches a target. For RPMD-GLE, repeat this for each distinct ring-polymer frequency. Equal-frequency partners share matrix parameters, but have separate auxiliary states and noise.
Actual offline GLEfit examples. P = 8, T = 300 K, η = 0.05 fs⁻¹ and ωc = 0.20 fs⁻¹, with an exponential bath spectrum. These saved examples are independent of the controls above.
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Inspect the fitted A matrix
The fit is performed before a simulation. Temperature and bead count determine the free ring-polymer frequencies; no molecular Hessian is needed. A new temperature or bead count generally requires a corresponding set of fits. Position-dependent atomistic coupling is an additional ingredient, and is not supplied by this scalar spectrum lesson.