Nonequilibrium Nonlocal Monte Carlo (NMC)

• Nonequilibrium Nonlocal Monte Carlo (NMC) is a family of algorithms that use driven, spatially inhomogeneous dynamics and nonlocal updates to enhance sampling and optimization.
• It deploys techniques like selective backbone heating, cluster updates, and finite-time switching to traverse complex energy landscapes in diverse systems.
• Recent advances incorporate reinforcement learning to dynamically select nonlocal moves, achieving significant speed-ups and improved solution quality compared to traditional methods.

Nonequilibrium Nonlocal Monte Carlo (NMC) denotes a family of Monte Carlo constructions in which the Markov dynamics are driven away from homogeneous equilibrium by spatially inhomogeneous temperatures, biased initial conditions, finite-time driving protocols, or explicitly nonlocal transport kernels. In the combinatorial-optimization literature, NMC refers most directly to algorithms that identify “backbone” variables in a low-energy basin and transiently heat only those subsets, thereby attempting to unfreeze rigid degrees of freedom without globally randomizing the configuration (Mohseni et al., 2021). Closely related work replaces heuristic backbone selection by a learned policy trained with deep reinforcement learning (Dobrynin et al., 14 Aug 2025). The same acronym also appears in cluster-update nonequilibrium-relaxation analyses of quantum criticality (Nonomura et al., 2019) and in particle-based simulations of nonlocal nonequilibrium transport, including ultrafast spin transport in Fe (Briones et al., 2021) and NEGF solvers with long-range polar-optical phonons (Ferry, 2023). A related but distinct development, Nonequilibrium Candidate Monte Carlo (NCMC), uses finite-time nonequilibrium proposals and work-based acceptance rules to preserve an equilibrium target distribution (Nilmeier et al., 2011).

1. Terminological scope and common structure

The literature uses the label “NMC” for several technically different algorithms. What they share is not a single update rule, but the deliberate use of nonequilibrium evolution together with a nonlocal mechanism for changing the state space explored by the sampler.

Usage	System class	Defining mechanism
NMC for optimization	random $k$-SAT, QAP	backbone-dependent $T_{\rm high}/T_{\rm low}$ updates
RLNMC	random and scale-free random 4-SAT	learned nonlocal transition policy
NER with loop QMC	quantum spin systems	cluster/loop updates and early-time relaxation
Kinetic NMC in Fe	ultrafast spin and charge transport	ballistic free flights plus scattering
NMC for NEGF	polar-optical phonon transport	Airy-kernel spatial jumps
NCMC	equilibrium molecular simulation	finite-time switching with work-based acceptance

Across these usages, “nonlocal” has different operational meanings. In optimization, the nonlocal move is a collective excitation of a learned backbone subset rather than a single-spin flip (Mohseni et al., 2021). In cluster-update QMC, an MCS builds all space-time loops and flips each loop with probability $1/2$, so the elementary update is a loop rather than a local plaquette change (Nonomura et al., 2019). In the Fe transport model, an electron’s post-scattering direction and speed determine a ballistic path that may span nanometers to tens of nanometers before the next event (Briones et al., 2021). In the NEGF formulation with polar-optical phonons, the Airy-function kernels define a stochastic spatial jump $s \to s'$ associated with a long-range Coulombic scattering event (Ferry, 2023).

The nonequilibrium element is likewise context-dependent. In optimization NMC, different subsets of variables experience different, time-dependent temperature profiles (Dobrynin et al., 14 Aug 2025). In nonequilibrium-relaxation QMC, independent runs are started from a biased initial condition and the early-time relaxation of an order parameter is analyzed before equilibration (Nonomura et al., 2019). In ultrafast transport, the initial electron ensemble is created by a femtosecond laser pulse and evolves under elastic and inelastic scattering before approaching diffusive behavior (Briones et al., 2021). In NCMC, candidate configurations are generated through a finite-time process in which the system is actively driven out of equilibrium, after which an acceptance criterion preserves the equilibrium distribution (Nilmeier et al., 2011).

2. Backbone heating and spatially inhomogeneous fluctuations in combinatorial optimization

In the formulation introduced by Mohseni et al., NMC is designed for discrete “spin-glass” Hamiltonians of the form

$$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$

with $s_i \in \{-1,+1\}$. Standard Parallel Tempering is taken as the baseline sampler, and NMC accelerates barrier crossing by intermittently applying a nonequilibrium, spatially inhomogeneous Markov chain that gives “extra heat” to learned “backbone” subsets of spins (Mohseni et al., 2021).

The geometric information used to identify those backbones is extracted from a surrogate Hamiltonian centered on a low-energy seed $s^*$,

$$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$

where $\epsilon_i=|h_i|+\sum_j |J_{ij}|$. Loopy Belief Propagation is then used to approximate the marginals of $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$. For pairwise terms $T_{\rm high}/T_{\rm low}$0, the cavity updates are

$T_{\rm high}/T_{\rm low}$1

$T_{\rm high}/T_{\rm low}$2

From these marginals one defines effective couplings

$T_{\rm high}/T_{\rm low}$3

and grows connected clusters by thresholding correlations. This learning stage is explicitly instance-wise: the backbone is not fixed a priori, but inferred from the local geometry of the current basin (Mohseni et al., 2021).

Once a backbone $T_{\rm high}/T_{\rm low}$4 has been selected, the algorithm applies a two-temperature Metropolis dynamics,

$T_{\rm high}/T_{\rm low}$5

with acceptance

$T_{\rm high}/T_{\rm low}$6

Because $T_{\rm high}/T_{\rm low}$7, barriers internal to $T_{\rm high}/T_{\rm low}$8 are flattened while the complement remains cold. The paper further states $T_{\rm high}/T_{\rm low}$9, and allows multi-level spatial profiles by binning $1/2$0-values into quantiles. To avoid bias, every inhomogeneous cycle is followed by $1/2$1 sweeps of standard homogeneous PT or Metropolis at $1/2$2 over the full system, a step termed “unlearning” and global equilibration (Mohseni et al., 2021).

The empirical results reported for this backbone-based NMC are specific and large. On random 4-SAT near $1/2$3 with $1/2$4, Survey Propagation solves only $1/2$5 of instances to within $1/2$6 approximation in 1 run, Adaptive PT solves $1/2$7 in $1/2$8 sweeps (4 repetitions), and NMC solves $1/2$9 in the same budget; for the hardest $s \to s'$0, NMC’s violations are one order of magnitude lower than SP’s best. Instance-wise, NMC is $s \to s'$1 APT on $s \to s'$2 of instances and $s \to s'$3 SP on $s \to s'$4, while whitening analysis finds that $s \to s'$5 of NMC’s best solutions contain $s \to s'$6 frozen clusters inaccessible to APT even after $s \to s'$7 repeats. On QAPLIB instances, NMC reaches $s \to s'$8 approximation in $s \to s'$9 sweeps for “Esc32a” and “Tho40”, versus APT’s $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$0 sweeps, and even NMC’s worst run outperforms APT’s best runs. The paper summarizes the time-to-solution gain as up to $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$1 faster than APT and $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$2 faster than SP for $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$3 quality (Mohseni et al., 2021).

3. Reinforcement-learned nonlocal policies

The reinforcement-learning extension of NMC keeps the same central premise—targeted high-temperature excitation of rigid variables—but replaces phenomenological backbone selection by a learned stochastic policy (Dobrynin et al., 14 Aug 2025). Conventional simulated annealing and parallel tempering are described there as homogeneous-temperature MCMC methods, whereas NMC operates in a nonequilibrium regime because different subsets of variables experience different, time-dependent temperature profiles. The intent is to unfreeze rigid regions of the landscape without fully randomizing the entire configuration, which would correspond to a homogeneous high-$$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$4 restart (Dobrynin et al., 14 Aug 2025).

In the original NMC construction, the backbone $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$5 can be chosen from local magnetizations or correlations,

$$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$6

or, in the simplified JAX implementation, from the instantaneous “make-break” field

$$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$7

A single nonlocal step consists of three stages: randomize or “hot-start” all spins in $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$8, hold $$H(s) = -\sum_i h_i s_i - \sum_{(ij)\in E} J_{ij} s_i s_j - \cdots - \sum_{a\in F} J_a \prod_{i\in \partial a} s_i,$$9 fixed and perform one sweep of Metropolis at $s_i \in \{-1,+1\}$0 on $s_i \in \{-1,+1\}$1, then release all spins and perform $s_i \in \{-1,+1\}$2 full sweeps at $s_i \in \{-1,+1\}$3. This kernel allows jumps of Hamming distance $s_i \in \{-1,+1\}$4 and typically costs $s_i \in \{-1,+1\}$5 ordinary MCMC sweeps plus two extra partial sweeps (Dobrynin et al., 14 Aug 2025).

RLNMC replaces the threshold rule by a policy

$s_i \in \{-1,+1\}$6

where $s_i \in \{-1,+1\}$7 indicates that spin $s_i \in \{-1,+1\}$8 is in the backbone at that step. The state $s_i \in \{-1,+1\}$9 consists of per-spin features—current local minimum spin $s^*$0, absolute local field $s^*$1, and recurrent memory $s^*$2—together with global features—best energy so far $s^*$3, current temperature $s^*$4, and global memory $s^*$5. The reward is

$s^*$6

The policy is parameterized by a factor-graph GNN with self-attention on each clause or hyperedge plus node aggregation, a per-node GRU, a global GRU, and MLP heads that produce $s^*$7 and a value estimate $s^*$8. Training uses Proximal Policy Optimization, and during training the paper runs $s^*$9 replicas in parallel, collecting trajectories $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$0 and updating $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$1 via PPO every few $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$2-steps (Dobrynin et al., 14 Aug 2025).

The reported benchmark regime is random 4-SAT. For scale-free 4-SAT with $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$3, $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$4, and 320 test instances, and for uniform-random 4-SAT with $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$5, $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$6, and 320 test instances, the metrics are $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$7, residual energy $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$8, and diversity $$H_\epsilon(r)=H(r)-\lambda \sum_i \epsilon_i s_i^* r_i,$$9. On scale-free 4-SAT, RLNMC cuts $\epsilon_i=|h_i|+\sum_j |J_{ij}|$0 by $\epsilon_i=|h_i|+\sum_j |J_{ij}|$1 in MC-sweeps and $\epsilon_i=|h_i|+\sum_j |J_{ij}|$2 in wall-time versus SA; NMC alone flattens TTS but does not attain RLNMC’s level. On uniform-random instances up to $\epsilon_i=|h_i|+\sum_j |J_{ij}|$3, RLNMC trained at $\epsilon_i=|h_i|+\sum_j |J_{ij}|$4 still improves over SA by $\epsilon_i=|h_i|+\sum_j |J_{ij}|$5–$\epsilon_i=|h_i|+\sum_j |J_{ij}|$6 in residual energy without re-tuning, whereas NMC without RL does not generalize as well. For the top $\epsilon_i=|h_i|+\sum_j |J_{ij}|$7 hardest instances, RLNMC yields a median diversity $\epsilon_i=|h_i|+\sum_j |J_{ij}|$8 higher than SA and $\epsilon_i=|h_i|+\sum_j |J_{ij}|$9 above NMC. The paper interprets this advantage through the overlap-gap-property: RLNMC learns intermediate-sized clusters that traverse “horizontal” distances in configuration space and dynamically adapts cluster sizes and excitation energies as the annealing temperature drops (Dobrynin et al., 14 Aug 2025).

4. Nonequilibrium relaxation and loop nonlocality in quantum Monte Carlo

A distinct usage of nonequilibrium nonlocal Monte Carlo arises in the nonequilibrium-relaxation approach of Nonomura and Tomita for quantum phase transitions (Nonomura et al., 2019). Here the key object is not a two-temperature backbone excitation, but the early-time relaxation of an order parameter under a cluster-update quantum Monte Carlo dynamics. The protocol replaces long-time equilibrium sampling by independent runs, termed random-number-sequence averages, each started from a biased initial condition. In a cluster or loop update QMC, one MCS consists of building all space-time loops in the current path-integral configuration and flipping each loop with probability $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$0. Simulation “time” $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$1 is simply the number of such MCSs elapsed (Nonomura et al., 2019).

The model used in the cited work is the 2D $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$2 columnar-dimer antiferromagnetic Heisenberg Hamiltonian

$\mu(r)\propto \exp[-\beta H_\epsilon(r)]$3

After the usual sublattice spin rotation to eliminate the sign problem, the simulation is carried out in the continuous-time path-integral representation. A disordered start can be chosen as the isolated-dimer product state $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$4, yielding $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$5, while an ordered start can be the classical Néel product state (Nonomura et al., 2019).

At the quantum critical point $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$6, the absolute value of the staggered magnetization measured on an initial Trotter layer,

$\mu(r)\propto \exp[-\beta H_\epsilon(r)]$7

obeys a stretched-exponential early-time form rather than a power law: $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$8 with $\mu(r)\propto \exp[-\beta H_\epsilon(r)]$9. The factor $T_{\rm high}/T_{\rm low}$00 is associated with “random-walk” cluster size growth, while $T_{\rm high}/T_{\rm low}$01 is the relaxation exponent and $T_{\rm high}/T_{\rm low}$02 a nonuniversal amplitude. The staggered susceptibility $T_{\rm high}/T_{\rm low}$03 grows as $T_{\rm high}/T_{\rm low}$04 with the same $T_{\rm high}/T_{\rm low}$05. Matching the nonequilibrium form to equilibrium finite-size scaling $T_{\rm high}/T_{\rm low}$06 yields

$T_{\rm high}/T_{\rm low}$07

which is the basis of the nonequilibrium-to-equilibrium data collapse used to locate $T_{\rm high}/T_{\rm low}$08 and extract $T_{\rm high}/T_{\rm low}$09 and $T_{\rm high}/T_{\rm low}$10 (Nonomura et al., 2019).

This formulation changes the role of “nonequilibrium” relative to optimization NMC. The simulation is not driven by spatially inhomogeneous temperatures; instead, the analysis is based on biased initial conditions and the functional form of the early-time relaxation. The nonlocality resides in the loop update itself. The cited work contrasts this with local-update world-line dynamics, where the critical decay obeys $T_{\rm high}/T_{\rm low}$11 and critical slowing down is described by a dynamical exponent $T_{\rm high}/T_{\rm low}$12. For cluster updates, the “power-law” stage disappears and is replaced by the stretched exponential, with no single $T_{\rm high}/T_{\rm low}$13 to describe the dynamics (Nonomura et al., 2019).

5. Particle-based NMC in nonequilibrium transport

In ultrafast transport and in stochastic formulations of NEGF, NMC denotes particle-based simulations in which the state evolution is both nonequilibrium and spatially nonlocal. The Fe study models spin-dependent dynamics after femtosecond laser excitation using a kinetic Monte Carlo scheme in a one-dimensional depth domain $T_{\rm high}/T_{\rm low}$14 with $T_{\rm high}/T_{\rm low}$15, assuming lateral homogeneity in $T_{\rm high}/T_{\rm low}$16. Each electron alternates between ballistic free flights and instantaneous scattering events. The free-flight time is sampled from

$T_{\rm high}/T_{\rm low}$17

and the velocity is $T_{\rm high}/T_{\rm low}$18 along the unit vector chosen at the previous scattering. The path-length distribution is

$T_{\rm high}/T_{\rm low}$19

The nonequilibrium initial condition comes from a laser pulse with photon energy $T_{\rm high}/T_{\rm low}$20 and FWHM $T_{\rm high}/T_{\rm low}$21, with positions sampled from the Beer–Lambert law $T_{\rm high}/T_{\rm low}$22, $T_{\rm high}/T_{\rm low}$23, and energies sampled from the spin-resolved occupied DOS of Fe. Elastic scattering is modeled as electron–lattice scattering with constant $T_{\rm high}/T_{\rm low}$24 or $T_{\rm high}/T_{\rm low}$25, and angular redistribution is sampled from the differential Mott cross section; inelastic electron–electron scattering generates secondary electrons using an impact-ionization approximation with $T_{\rm high}/T_{\rm low}$26, spin-resolved DOS sampling, and a spin-flip probability $T_{\rm high}/T_{\rm low}$27 from Hong and Mills. At both $T_{\rm high}/T_{\rm low}$28 and $T_{\rm high}/T_{\rm low}$29, open boundaries remove exiting electrons. Simulations use $T_{\rm high}/T_{\rm low}$30 trajectories up to $T_{\rm high}/T_{\rm low}$31. The reported physical insight is that shorter $T_{\rm high}/T_{\rm low}$32 leads to larger spatial spread, slower net forward velocity, and earlier onset of the diffusive regime, while secondary-electron generation prolongs the superdiffusive regime, extends the temporal profile of $T_{\rm high}/T_{\rm low}$33 at $T_{\rm high}/T_{\rm low}$34, and increases the integrated spin flux (Briones et al., 2021).

The NEGF application treats long-range polar-optical phonon scattering in mixed $T_{\rm high}/T_{\rm low}$35 space after an Airy transform in $T_{\rm high}/T_{\rm low}$36. The local distribution is extracted from the Keldysh Green function through

$T_{\rm high}/T_{\rm low}$37

and in practice by solving the integral equation

$T_{\rm high}/T_{\rm low}$38

Polar-optical phonon scattering enters through the Fröhlich matrix element

$T_{\rm high}/T_{\rm low}$39

and the retarded and lesser self-energies contain Airy-function double integrals. In the Monte Carlo picture, those double integrals are interpreted as a spatial jump distribution, with $T_{\rm high}/T_{\rm low}$40 sampled from a normalized Airy kernel and momentum transfer sampled from the angular integral $T_{\rm high}/T_{\rm low}$41. The algorithm computes the total rate $T_{\rm high}/T_{\rm low}$42, samples the free-flight length, propagates in Airy coordinate under a uniform electric field, then samples phonon emission or absorption, momentum transfer $T_{\rm high}/T_{\rm low}$43, and the new Airy coordinate $T_{\rm high}/T_{\rm low}$44; after each global sweep the self-energies are recomputed from the current $T_{\rm high}/T_{\rm low}$45. The paper states lowest-order self-consistent Born approximation, open contacts with Maxwellian injection, $T_{\rm high}/T_{\rm low}$46–$T_{\rm high}/T_{\rm low}$47, and $T_{\rm high}/T_{\rm low}$48 global cycles to converge. It further reports robust convergence of the velocity–field curve up to $T_{\rm high}/T_{\rm low}$49, $T_{\rm high}/T_{\rm low}$50–$T_{\rm high}/T_{\rm low}$51 desktop runtimes, and minor $T_{\rm high}/T_{\rm low}$52 uncertainties in high-field velocity from boundary conditions and energy discretization (Ferry, 2023).

These transport formulations show that, outside optimization, NMC can refer to stochastic solvers in which the nonlocality is literal in real space: electrons propagate over nanometric free flights or undergo Airy-kernel spatial jumps rather than executing local updates on an abstract combinatorial graph.

6. Relation to Nonequilibrium Candidate Monte Carlo

Nonequilibrium Candidate Monte Carlo is not the same construction as backbone-based NMC, but it provides a closely related and formally exact equilibrium framework for nonequilibrium proposals (Nilmeier et al., 2011). The target distribution is

$T_{\rm high}/T_{\rm low}$53

and a candidate configuration is generated not by a single perturbation, but by a finite-time switching protocol

$T_{\rm high}/T_{\rm low}$54

where $T_{\rm high}/T_{\rm low}$55 are perturbation kernels and $T_{\rm high}/T_{\rm low}$56 are propagation kernels. Along the resulting trajectory, the nonequilibrium work and heat are

$T_{\rm high}/T_{\rm low}$57

with $T_{\rm high}/T_{\rm low}$58. Pathwise detailed balance leads to a Metropolis–Hastings acceptance ratio involving protocol probabilities, perturbation kernels, path actions, and the reduced-potential change. For symmetric single-state driving with detailed-balance MCMC relaxation,

$T_{\rm high}/T_{\rm low}$59

and for $T_{\rm high}/T_{\rm low}$60 with identity relaxation, this reduces exactly to the standard Metropolis criterion (Nilmeier et al., 2011).

The practical motivation is similar to that of other nonequilibrium Monte Carlo constructions: avoid proposals that are either too local to decorrelate the chain or so large that their acceptance becomes negligible. Nilmeier et al. illustrate this with a bistable dimer in dense WCA solvent. In vacuum, 500 MD steps alone give a dimer-extension autocorrelation $T_{\rm high}/T_{\rm low}$61 iterations, and adding one instantaneous MC step reduces $T_{\rm high}/T_{\rm low}$62. In dense solvent, 500 MD steps alone give $T_{\rm high}/T_{\rm low}$63, while instantaneous $T_{\rm high}/T_{\rm low}$64 moves still leave $T_{\rm high}/T_{\rm low}$65 because acceptance is virtually zero. By contrast, a 2048-step NCMC move achieves acceptance $T_{\rm high}/T_{\rm low}$66, giving $T_{\rm high}/T_{\rm low}$67 iterations. The cost rises from 500 to approximately 2548 force evaluations, but the net result is an overall $T_{\rm high}/T_{\rm low}$68 increase in production of uncorrelated samples. Acceptance rises superlinearly with switching length for $T_{\rm high}/T_{\rm low}$69, plateaus around 2048–8192 steps at $T_{\rm high}/T_{\rm low}$70–$T_{\rm high}/T_{\rm low}$71, and beyond $T_{\rm high}/T_{\rm low}$72 steps the additional cost outweighs the marginal gain (Nilmeier et al., 2011).

The conceptual boundary is therefore sharp. In optimization NMC, nonequilibrium and nonlocality are used to accelerate search by selectively heating learned subsets, and the objective is low-energy optimization rather than exact equilibrium sampling (Mohseni et al., 2021). In cluster-update nonequilibrium-relaxation QMC, the nonequilibrium component is the biased start and early-time analysis, while the nonlocality is the loop update (Nonomura et al., 2019). In transport NMC, the Monte Carlo particles directly realize nonlocal motion in physical space (Briones et al., 2021, Ferry, 2023). In NCMC, by contrast, candidate generation is itself a nonequilibrium trajectory, but the acceptance rule is designed so that the equilibrium target remains stationary (Nilmeier et al., 2011). A plausible implication is that “Nonequilibrium Nonlocal Monte Carlo” is best understood as a methodological category rather than a single algorithm: the shared idea is to use driven, nonlocal dynamics where conventional homogeneous local updates become ineffective, but the invariant measure, observables, and notion of nonlocality are application-specific.