Monte Carlo framework for noncontinuous interactions between particles and classical fields

Particles and fields are standard components in numerical calculations like transport simulations in nuclear physics and have well-understood dynamics. Still, a common problem is the interaction between particles and fields due to their different formal description. Particle interactions are discrete, pointlike events while field dynamics is described with continuous partial-differential equations of motion. A workaround is the use of effective theories like the Langevin equation with the drawback of energy conservation violation. We present a method, which allows us to model noncontinuous interactions between particles and scalar fields, allowing us to simulate scattering-like interactions which exchange discrete “quanta” of energy and momentum between fields and particles while obeying energy and momentum conservation and allowing control over interaction strengths and times. In this paper we apply this method to different model systems, starting with a simple harmonic oscillator, which is damped by losing discrete energy quanta. The second and third system consists of an oscillator and a one-dimensional field, which are damped via discrete energy loss and are coupled to a stochastic force, leading to equilibrium states which correspond to statistical Langevin-like systems. The last example is a scalar field in (1 + 3) space-time dimensions, which is coupled to a microcanonical ensemble of particles by incorporating particle production and annihilation processes. Obeying the detailed-balance principle, the system equilibrates to thermal and chemical equilibrium with dynamical fluctuations on the fields, generated dynamically by the discrete interactions.

Figure 1

Visualization of the interaction principle, as described in Eq. (12). The energy of the initial field $\varphi \left(t\right)$ is changed by a parameterized field variation $\delta \varphi$. The resulting field $\varphi \left(t\right)+\delta \varphi$ is increased by a given energy $\Delta E$ and momentum $\Delta \mathit{P}$. The traveling direction of the Gaussian depends on the momentum.

Figure 2

Simulation of a discretely damped harmonic oscillator. The number of simulated particles and energy excitations is $N=150$. With every decay of a particle, the oscillator loses a bit of energy, which leads to damping of its motion. The discrete and continuous version have the same ensemble average. In a single run statistical fluctuations occur.

Figure 3

Simulation of a simple harmonic oscillator, coupled to a heat bath via a Langevin equation. The random force and the damping is implemented with the noncontinuous energy-transfer method. The plot shows the distribution function of the energy, the position distribution and the velocity distribution for a temperature of $T=0.5\mathrm{arb}.\mathrm{units}$, and ${10}^8$ calculation steps and a corresponding energy-step size. Both plots coincide with the expected analytical result, which is $exp(-E/T)$ for the energy distribution (top) and $1/\left(\sqrt{2\pi T}\right)exp(-m{\omega }_0^2{x}^2/2T)$ for the spatial distribution.

Figure 4

Time-averaged power spectrum of the scalar field $\varphi (x,t)$ in (1 + 1) space-time dimensions. The spectrum is calculated according to (43). The system is a one-dimensional harmonic oscillator, coupled to a Langevin equation with a continuous and a noncontinuous ansatz. The dashed blue line shows the theoretical solution of the system $T/k^2$, the red circles the noncontinuous approach, and the green diamonds show the classical Langevin equation as a reference. All three curves are in very good agreement in the intermediate region and show the same system temperature, which was chosen to $T=0.5\mathrm{arb}.\mathrm{units}$. Both Fourier plots are averaged over 100 snapshots of the same run. Simulated was a grid with 1024 points. At large wavelengths, both systems deviate due to finite-system-size effects. At small wavelengths the noncontinuous methods has an interesting cutoff due to the finite size of the interaction region in comparison to the pointlike interactions of the classical Langevin method.

Figure 5

Energy fluctuations of the scalar field and the particles, $E\left(t\right)-\langle E\rangle$. Field and particles exchange energy by particle production and annihilation processes. While the total energy of the system is conserved and shows only numerical fluctuations, the energy of the components show thermal fluctuations, which are anticorrelated due to total-energy conservation. The relative fluctuations of the field's energy is $\sim {10}^{-2}$ and of the quarks $\sim {10}^{-3}$. The total energy fluctuates on a scale of $\left|\Delta E\right|/\langle E\rangle \lesssim 5\times {10}^{-5}$.

Figure 6

Total particle number in the thermal-box simulation. Particles can annihilate, and their momentum and energy are transferred to the scalar field in form of scalar-field excitations. Because of the dynamic nature of this process, the total particle number fluctuates around the average thermal value.

Figure 7

Distribution of the scalar field, showing the expected thermal Gauss-distributed fluctuations. The mean of the Gaussian can drift slowly with time due to fluctuations of the scalar mean field. Particle annihilation increases the local fluctuations of the scalar field, and particle production damps them by dissipating energy from the interaction region.

Figure 8

Plot of the 3D field with a cut in the $x-y$ plane at different times in the simulation. The simulation starts with a uniform scalar field without any excited modes. Due to particle creation and annihilation, field fluctuations are created dynamically within the simulation. The color coding shows the value of the scalar field. Left: Some particles have annihilated and have created small, local excitations of the scalar field in form of moving, Gaussian-shaped blobs. Right: The equilibrated field in the long-time limit: Due to the particle-field interactions, Gaussian fluctuations are dynamically created by microscopic interactions.

Figure 9

Spectrum of the field's kinetic energy ${\dot{\sigma }}^2/2$: While a thermal spectrum of a classical field has an average energy of $kT$ for every mode due to the equipartion theorem, with $T=0.15\mathrm{arb}.\mathrm{units}$, the simulated field has an evenly distributed energy per mode only for large wavelengths. For small wavelengths, the spectrum is suppressed by a soft cutoff. This cutoff is due to the finte spatial extent of the interactions on the field. The deviation from the analytical result at higher modes can be explained by the nonlinear potential in the field equations of motion.

Figure 10

Distribution function of the particle energies showing the expected thermal Boltzmann distribution $f_q\left(E\right)\sim exp(-E/T)$. By microscopic interactions particles and field can exchange energy and momentum and equilibrate to the same temperature $T=0.15\mathrm{arb}.\mathrm{units}$ via annihilation and creation processes, demonstrating the proper implementation of the principle of detailed balance in our simulation.

Figure 11

Rapid expansion of a hot quark-droplet as a simply nonequilibrium setup for a heavy-ion collision. The interaction between quarks and fields are implemented with the method purposed in this paper. The chiral $\sigma$ field can remit latent energy, which is freed by the phase transition, by creating particles. Results of this study will be presented in an upcoming paper. Left: Quark density; right: mean field value of the $\sigma$ field.