The second order Langevin equation and numerical simulations

Horowitz, Alan M.

doi:10.1016/0550-3213(87)90159-3

Cited by 48 publications

(29 citation statements)

References 7 publications

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“…must coincide with the ordinary correlation functions of the fundamental field in momentum space [14][15][16]. In this section, we show that the two-and the four-point autocorrelation functions do have this property at one-loop order of perturbation theory.…”

Section: Relation To the Ordinary Correlation Functionsmentioning

confidence: 99%

“…Moreover, in the second-order form, 5) and after substituting t → 2µ 0 t, the evolution equations are seen to coincide with the Langevin equation up to a term that goes to zero at large µ 0 . Since its introduction by Horowitz [14][15][16], the generalized HMC algorithm has been occasionally studied in the literature, where it is referred to as the Kramers equation or the L2MC algorithm (see refs. [13,17,18], for example).…”

Section: Evolution Equationsmentioning

confidence: 99%

“…The generalized HMC algorithm [14][15][16] integrates a stochastic version of the molecular-dynamics 1 Equation (1.1) is expected to hold on the infinite lattice and on finite lattices with open boundary conditions [10]. On lattices with periodic boundary conditions, topology-changing tunneling transitions can give rise to very large autocorrelation times [7][8][9][10].…”

Section: Stochastic Molecular Dynamicsmentioning

confidence: 99%

“…As usual the molecular dynamics evolves the field φ(t, x) together with its momentum π(t, x) as a function of a fictitious time t. The stochastic evolution equations [14][15][16] …”

Section: Evolution Equationsmentioning

confidence: 99%

“…For simplicity the φ 4 theory is considered, but our main result (the non-renormalizability of the molecular-dynamics equations) no doubt extends to most theories of interest. A slightly generalized version of the HMC algorithm is studied, which was introduced many years ago by Horowitz [14][15][16] (see sections 2 and 3). The non-renormalizability of the associated stochastic equation is then established by showing that the four-point autocorrelation function of the fundamental field has a non-removable ultraviolet singularity at second order in the coupling (sections 4 and 5).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Non-renormalizability of the HMC algorithm

Lüscher

Schaefer

2011

J. High Energ. Phys.

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In lattice field theory, renormalizable simulation algorithms are attractive, because their scaling behaviour as a function of the lattice spacing is predictable. Algorithms implementing the Langevin equation, for example, are known to be renormalizable if the simulated theory is. In this paper we show that the situation is different in the case of the molecular-dynamics evolution on which the HMC algorithm is based. More precisely, studying the φ 4 theory, we find that the hyperbolic character of the molecular-dynamics equations leads to non-local (and thus non-removable) ultraviolet singularities already at one-loop order of perturbation theory.

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Section: Relation To the Ordinary Correlation Functionsmentioning

confidence: 99%

Section: Evolution Equationsmentioning

confidence: 99%

Section: Stochastic Molecular Dynamicsmentioning

confidence: 99%

“…As usual the molecular dynamics evolves the field φ(t, x) together with its momentum π(t, x) as a function of a fictitious time t. The stochastic evolution equations [14][15][16] …”

Section: Evolution Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-renormalizability of the HMC algorithm

Lüscher

Schaefer

2011

J. High Energ. Phys.

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The Theory of Hybrid Stochastic Algorithms

Kennedy

1990

Probabilistic Methods in Quantum Field Theory and Quantum Gravity

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These lectures introduce the family of Hybrid Stochastic Algorithms for performing Monte Carlo calculations in Quantum Field Theory. After explaining the basic concepts of Monte Carlo integration we discuss the properties of Markov processes and one particularly useful example of them: the Metropolis algorithm. Building upon this i&nework we consider the Hybrid and Lang&n algorithms from the viewpoint that they are appr oximate versions of the Hybrid Monte Carlo method; and thus we are led to consider Molecular Dynamics using the Leapfrog algorithm. The lectures conclude by reviewing recent progress in these areas, explaining higher-order integration schemes, the asymptotic large-volume belmviour of the various algorithms, and some simple exact results obtained by applying them to free field theory. It is attempted throughout to give simple yet correct proofs of the various results encountered.

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Lattice QCD without topology barriers

Lüscher

Schaefer

2011

J. High Energ. Phys.

248

250

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As the continuum limit is approached, lattice QCD simulations tend to get trapped in the topological charge sectors of field space and may consequently give biased results in practice. We propose to bypass this problem by imposing open (Neumann) boundary conditions on the gauge field in the time direction. The topological charge can then flow in and out of the lattice, while many properties of the theory (the hadron spectrum, for example) are not affected. Extensive simulations of the SU(3) gauge theory, using the HMC and the closely related SMD algorithm, confirm the absence of topology barriers if these boundary conditions are chosen. Moreover, the calculated autocorrelation times are found to scale approximately like the square of the inverse lattice spacing, thus supporting the conjecture that the HMC algorithm is in the universality class of the Langevin equation.Comment: Plain TeX source, 26 pages, 4 figures include

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The second order Langevin equation and numerical simulations

Cited by 48 publications

References 7 publications

Non-renormalizability of the HMC algorithm

Non-renormalizability of the HMC algorithm

The Theory of Hybrid Stochastic Algorithms

Lattice QCD without topology barriers

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