NVT ensemble¶

Nosé-Hoover thermostat¶

Unlike Berendsen or Langevin, the Nosé-Hoover thermostat doesn't bolt onto an existing integration scheme — it replaces the time-integration scheme itself, extending the equations of motion with an extra thermostat degree of freedom \(\eta\) that couples the system to a target temperature.

Nosé-Hoover thermostat equations

At initialization, the thermostat variable and its derivatives start at zero, and the coupling frequency is set from the user-provided coupling period \(t_{period}\) (Tdamp):

\[ \eta = \dot{\eta} = \ddot{\eta} = 0, \qquad t_{freq} = \frac{1}{t_{period}} \]

At each step, given the target temperature \(T^*\) (which may itself be constant, linearly ramped, or piecewise-interpolated over time — same three modes as the other thermostats), the target kinetic energy is:

\[ N_{dof} = 3 N_{atoms} - 3, \qquad KE^* = N_{dof} \, k_B \, T^* \]

and the thermostat's own mass and acceleration are:

\[ \eta_M = \frac{KE^*}{t_{freq}^2}, \qquad \ddot{\eta} = t_{freq}^2 \left( \frac{KE_{cur}}{KE^*} - 1 \right) \]

This feeds a velocity-scaling factor applied to every atom alongside the usual position/velocity integration steps, keeping the measured kinetic energy oscillating around \(KE^*\) rather than being rescaled to it exactly every step (unlike Berendsen).

Syntax

init_nose_hoover:
  algo: NVT
  Tstart: <float>
  Tend: <float>
  Tdamp: <float>
  tchain: <int>

Parameters

algo:    string, default "NVT"    # "NVT" here — "NPT" adds a barostat, see NPT ensemble.
Tstart:  float, required          # Target temperature at the start of the run.
Tend:    float, optional          # Target temperature at the end of the run (linear ramp); defaults to Tstart if omitted.
Tdamp:   float, default 0.1       # Thermostat coupling time.
tchain:  int, default 3           # Length of the Nosé-Hoover chain.

This is a genuinely different parameter set from Berendsen/Langevin — there's no T/Tstart+Tstop/tserie+Tserie three-way choice here, just Tstart (required) and an optional Tend for a linear ramp.

init_nose_hoover only sets up the thermostat context; the actual per-step integration is done by a handful of companion operators (setup_nose_hoover, nhc_temp_integrate, …) wired automatically once config_nose_hoover.msp is included and numerical_scheme points at verlet_nhnvt:

Usage example

includes:
  - config_nose_hoover.msp

init_nose_hoover:
  algo: NVT
  Tstart: 150. K
  Tend:   150. K
  Tdamp:  0.05 ps
  tchain: 3

numerical_scheme: verlet_nhnvt

Warning

config_nose_hoover.msp must be included — it's what wires the per-step Nosé-Hoover operators into +init_epilog/numerical_scheme. Without it, init_nose_hoover's output context isn't consumed by anything. Its full content (exaStamp/data/config/config_nose_hoover.msp):

+init_prolog:
  - deformation_xform:
      defbox: { extension: [ 1.0 , 1.0 , 1.0 ] }

+init_epilog:
  - init_nose_hoover

nose_hoover_additional_step:
  - setup_nose_hoover
  - couple_npt

The `verlet_nhnvt` scheme¶

verlet_nhnvt (exaStamp/data/config/config_numerical_schemes.msp) expands to:

Usage example

verlet_nhnvt:
  name: NHNVT_scheme
  body:
    - simulation_thermodynamic_state
    - nhc_temp_integrate
    - nh_v_temp:
        rebind: { value: vscale }
        body: [ scale_v ]
    - compute_vel_bias:
        rebind: { out: vbias }
        body: [ avg_v_m ]
    - remove_vel_bias:
        rebind: { value: vbias }
        body: [ shift_v ]
    - push_f_v: { dt_scale: 0.5, xform_mode: IDENTITY }
    - push_v_r: { dt_scale: 1.0, xform_mode: INV_XFORM }
    - check_and_update_particles
    - load_balance_auto_tune_start
    - compute_all_forces_energy
    - push_f_v: { dt_scale: 0.5, xform_mode: IDENTITY }
    - simulation_thermodynamic_state
    - nhc_temp_integrate
    - nh_v_temp:
        rebind: { value: vscale }
        body: [ scale_v ]
    - load_balance_auto_tune_end

Unlike the plain-Verlet schemes, this isn't verlet_first_half/verlet_second_half plus a bolted-on thermostat step — the thermostat integration is interleaved with the position/velocity pushes on both half-steps, LAMMPS-fix nvt-style:

nhc_temp_integrate (exaStamp/src/npt/nhc_temp_integrate.cpp) advances the Nosé-Hoover chain (length tchain) from the current kinetic energy and target KE^*, and produces a uniform velocity-scaling factor on its vscale output.
nh_v_temp isn't a standalone operator — it's a named batch that runs scale_v (exaNBody/src/compute/generic_op_vec3.cu) with its value input slot rebound to vscale, i.e. it multiplies every particle's velocity by the chain's scaling factor.
compute_vel_bias batches avg_v_m (exaStamp/src/compute/avg_v_m.cu), rebinding its out output to vbias — a mass-weighted average velocity of the whole system, negated.
remove_vel_bias batches shift_v, rebinding its value input to vbias — adding that negated average to every particle's velocity, which cancels out the system's net (center-of-mass) drift after the thermostat scaling.
The thermostat/bias block runs twice per step (once before the position push, once after the force recomputation) — the same Nosé-Hoover chain half-step pattern used by verlet_nhnpt below.

rebind/body here is the same generic ONIKA batch mechanism as numerical_scheme itself: a named block (nh_v_temp, compute_vel_bias, remove_vel_bias) that wraps one or more operators and renames one of their slots for that particular use, rather than a distinct operator type.

NVT ensemble¶

Nosé-Hoover thermostat¶

The verlet_nhnvt scheme¶

The `verlet_nhnvt` scheme¶