Relativistic and Quantum Revisions of the Multisource Thermal Model in High-Energy Collisions

Abstract

The multisource thermal model is revised for relativistic and quantum situations. It is shown that the quantum effect can be neglected due to a very small result. The distributions of particle momenta, momentum components, transverse momenta, kinetic energies, and velocities in both the classical and relativistic situations are presented to give comparisons.

1. Introduction

The multisource thermal model (or previously the multisource ideal gas model) was suggested by us in the past years [1–4]. This model assumes that many emission sources of produced particles and nuclear fragments are formed in high-energy collisions. Each emission source is treated as a thermal equilibrium system of classical ideal gas. Then, the classical Maxwell’s ideal gas model and corresponding distributions [5] can be used in the multisource thermal model. Because the classical Maxwell’s ideal gas model has no consideration of the relativistic and quantum effects, some revisions are needed when we consider the model’s application in high-energy collisions.

In this paper, the relativistic and quantum effects are considered for each emission source. The multisource thermal model [1–4] is then revised for the relativistic and quantum situations. Distributions of particle momenta, momentum components, transverse momenta, kinetic energies, and velocities in both the classical and relativistic situations are given and compared.

2. Formulations and Comparisons

Let 𝑘 and 𝑇 denote the Boltzmann constant and temperature of thermal equilibrium system, respectively, and 𝑚₀, 𝑝, 𝑝_{𝑥,𝑦,𝑧}, 𝑝_𝑇, 𝐸_𝐾, 𝑣, 𝜃, and 𝜑 denote the rest mass, momentum, momentum components, transverse momentum, kinetic energy, velocity, emission angle, and azimuth of a concerned particle, respectively. In our discussions, we define the beam direction to be the 𝑜𝑧 axis and the reaction plane to be the 𝑥𝑜𝑧 plane in high-energy collisions.

In the classical Maxwell’s ideal gas model [5], we have the distributions of particle momenta, momentum components, transverse momenta, kinetic energies, and velocities to be 𝑓_𝑝𝑝=1𝑁𝑑𝑁=𝑑𝑝2𝜋𝑝²𝑚₀𝑘𝑇^3/2−𝑝exp²2𝑚₀,𝑓𝑘𝑇_{𝑝𝑥,𝑦,𝑧}𝑝_{𝑥,𝑦,𝑧}=1𝑁𝑑𝑁𝑑𝑝_{𝑥,𝑦,𝑧}=1√2𝜋𝑚₀−𝑝𝑘𝑇exp^{2𝑥,𝑦,𝑧}2𝑚₀,𝑓𝑘𝑇_𝑝𝑇𝑝_𝑇=1𝑁𝑑𝑁𝑑𝑝_𝑇=𝑝_𝑇𝑚₀−𝑝𝑘𝑇exp^2𝑇2𝑚₀,𝑓𝑘𝑇_𝐸𝐾𝐸_𝐾=1𝑁𝑑𝑁𝑑𝐸_𝐾=2√𝜋1𝑘𝑇^3/2𝐸_𝐾−𝐸exp^𝐾,𝑓𝑘𝑇_𝑣(𝑣)=1𝑁𝑑𝑁=𝑑𝑣2𝜋𝑚₀𝑘𝑇^3/2𝑣²−𝑚exp⁰𝑣²,2𝑘𝑇 (2.1)(2.2)(2.3)(2.4)(2.5) respectively.

Considering the relativistic effect [6, 7], we have 𝑓_𝑝𝑝=1𝑁𝑑𝑁𝑑𝑝=𝐶𝑝²exp−𝑝²+𝑚²₀𝑘𝑇, (2.6) where 𝐶=(1/𝑚²₀𝑘𝑇)(1/𝐾₂(𝑚₀/𝑘𝑇)) is the normalization constant and 𝐾₂(𝑚₀/𝑘𝑇) is the modified Bessel function of order 2. In the above equation, we have taken the natural unit system in which the speed 𝑐 of light in vacuum is 1. Considering the quantum effect [8], we have 𝑓_𝑝𝑝=1𝑁𝑑𝑁𝑑𝑝=𝐶₁𝑝²exp𝑝²+𝑚²₀−𝜇𝑘𝑇±1⁻¹, (2.7) where 𝐶₁ is the normalization constant, 𝜇 is the chemical potential, +1 denotes fermions, and −1 denotes bosons, respectively. Our calculations show that, comparing with (2.6), the quantum effect in (2.7) with 𝜇 having large dispersion from 𝑚₀ is so small that we do not need to consider it in studying momentum distributions in high-energy collisions. The chemical potential also causes a very small effect in the case of neglecting ±1. Only the combination of quantum effect and 𝜇≈𝑚₀ causes an obvious dispersion from (2.6). As an example, by taking 𝑚₀=938MeV/𝑐² and 𝜇=937MeV, the calculated momentum distributions for relativistic nonquantum gas (see (2.6)), boson gas (see (2.7) with −1), and fermion gas (see (2.7) with +1) at 𝑘𝑇=10,20,50, and 100?MeV are shown in Figure 1 by the solid, dotted, and dashed curves, respectively. One can see obvious differences in momentum distributions among the three kinds of ideal gases. Especially at higher temperature, this difference is more obvious.

Figure 1 

Momentum distributions of relativistic nonquantum particles (solid curves), bosons (dotted curves), and fermions (dashed curves) in ideal gas model calculated by (2.6) and (2.7), respectively, with 𝑚0=938MeV/𝑐2 and 𝜇=937MeV. The results are given for four temperatures indicated in the figure.

As the relativistic and quantum revisions of (2.1), (2.7) can be used in the multisource thermal model. To avoid the unbending choice of 𝜇 and ±1, in the following we use (2.6) to give comparisons with (2.1)–(2.5). In Figure 2, the particle momentum distributions at 𝑘𝑇=10,20,50, and 100?MeV calculated by (2.6) and (2.1) are given by the solid and dashed curves, respectively. In the calculation, we take 𝑚₀=938MeV/𝑐² which means that an ideal proton gas is considered. One can see that the difference between the classical distribution and relativistic one is not obvious at lower temperatures (10 and 20?MeV), while the difference is obvious at higher temperatures (50 and 100?MeV).

Figure 2 

Proton momentum distributions in relativistic (nonquantum) ideal gas model (solid curves) and classical ideal gas model (dashed curves). The different curves are calculated by (2.6) and (2.1), respectively, and are given for four temperatures indicated in the figure. The solid curves in Figure 2 are the same as those in Figure 1.

The momentum components can be given by 𝑝_𝑥=𝑝sin𝜃cos𝜑, 𝑝_𝑦=𝑝sin𝜃sin𝜑, and 𝑝_𝑧=𝑝cos𝜃, respectively. An isotropic ideal gas model gives the same distribution of three different components, and the emission angular and azimuthal distributions are 𝑓_𝜃(𝜃)=(1/2)sin𝜃 and 𝑓_𝜑(𝜑)=1/2𝜋, respectively. Combining with (2.6), we can use the Monte Carlo method and analytical expression to give the momentum component distributions at relativistic energy. In Figure 3, the dashed curve represents the classical distribution (see (2.2)) of 𝑝_𝑥, and the dotted and solid curves represent the relativistic distributions by using the Monte Carlo method and analytical expression 𝑓_𝑝𝑥𝑝_𝑥=1𝑁𝑑𝑁𝑑𝑝_𝑥=𝐶₂𝑝²_𝑥+𝑚²₀exp−𝑝²_𝑥+𝑚²₀𝑘𝑇, (2.8) respectively, where 𝐶₂ is the normalization constant. All the distributions are given at four temperatures mentioned in the figure. We see that the difference between the classical and relativistic distributions at lower temperature is not obvious, and the difference at higher temperature is obvious. The relativistic distributions obtained by the Monte Carlo method and analytical expression are almost the same. This result confirms the two methods all together.

Figure 3 

Proton momentum component distributions in relativistic ideal gas model (solid and dotted curves) and classical ideal gas model (dashed curves). The different curves are calculated by (2.8), Monte Carlo method, and (2.2), respectively, and are given for four temperatures indicated in the figure.

The transverse momentum is given by 𝑝_𝑇=𝑝sin𝜃. Combining with (2.6), we can use the Monte Carlo method and analytical expression to give the transverse momentum distribution at relativistic energy. In Figure 4, the dashed curve represents the classical distribution (see (2.3)) of 𝑝_𝑇, whereas the dotted and solid curves represent the relativistic results by using the Monte Carlo method and Boltzmann distribution 𝑓_𝑝𝑇𝑝_𝑇=1𝑁𝑑𝑁𝑑𝑝_𝑇=𝐶₃𝑝_𝑇exp−𝑝²_𝑇+𝑚²₀𝑘𝑇, (2.9) respectively, where 𝐶₃ is the normalization constant. All the distributions are given at four temperatures. We see again that the difference between the classical distribution and relativistic one is small at lower temperature and large at higher temperature. The relativistic results obtained by the Monte Carlo method and Boltzmann distribution are almost the same. This confirms the two methods all together.

Figure 4 

Proton transverse momentum distributions in relativistic ideal gas model (solid and dotted curves) and classical ideal gas model (dashed curves). The different curves are calculated by (2.9), Monte Carlo method, and (2.3), respectively, and are given for four temperatures indicated in the figure.

The kinetic energy in relativistic situation can be given by 𝐸_𝐾=𝑝²+𝑚²₀−𝑚₀; combining with (2.6), we have the distribution of kinetic energy in relativistic situation to be 𝑓_𝐸𝐾𝐸_𝐾=1𝑁𝑑𝑁𝑑𝐸_𝐾=𝐶𝐸_𝐾+𝑚₀𝐸_𝐾+𝑚₀²−𝑚²₀−𝐸exp^𝐾+𝑚₀.𝑘𝑇 (2.10) The total energy 𝐸=𝑝²+𝑚²₀=𝐸_𝐾+𝑚₀ is simply given by 𝑓_𝐸(𝐸)=1𝑁𝑑𝑁𝑑𝐸=𝐶𝐸𝐸²−𝑚²₀−𝐸exp.𝑘𝑇 (2.11) For ideal gas in classical situation, the energy is only the kinetic energy due to zero potential energy. The comparisons between the kinetic energy distributions in classical situation (dashed curves) and relativistic case (solid curves) are presented in Figure 5. The two curves at lower temperature are almost the same, and those at higher temperature are very similar.

Figure 5 

Proton kinetic energy distributions in relativistic ideal gas model (solid curves) and classical ideal gas model (dashed curves). The different curves are calculated by (2.10) and (2.4), respectively, and are given for four temperatures indicated in the figure.

The relationship between momentum and velocity at relativistic energy gives 𝑝=𝑚₀𝑣/√1−𝑣². Combining with (2.6), we have the distribution of velocity to be 𝑓_𝑣(𝑣)=1𝑁𝑑𝑁𝑚𝑑𝑣=𝐶³⁰𝑣²1−𝑣^25/2−𝑚exp⁰𝑘𝑇√1−𝑣². (2.12) The comparisons between the velocity distributions in classical situation (dashed curves) and relativistic case (solid curves) are presented in Figure 6. There are small differences at lower temperatures and large differences at higher temperatures. One sees 𝑣>𝑐 in classical situation at around 100?MeV. Especially, the probability of particles with 𝑣>𝑐 is very large at 400?MeV. Obviously, the relativistic effect has to be considered at these higher temperatures.

Figure 6 

Proton velocity distributions in relativistic ideal gas model (solid curves) and classical ideal gas model (dashed curves). The different curves are calculated by (2.12) and (2.5), respectively, and are given for five temperatures indicated in the figure.

3. Conclusion

In conclusion, the multisource thermal model [1–4] is revised in this paper for relativistic and quantum situations. The quantum effect in particle momentum distributions in high-energy collisions is very small when the chemical potential has a large dispersion from the rest mass of the concerned particle. The chemical potential also causes a very small effect when we neglect ±1. Only the combination of quantum effect and nearly the same chemical potential and rest mass causes an obvious dispersion from the classical distribution. In the distributions of particle momenta, momentum components, transverse momenta, kinetic energies, and velocities, the relativistic effect is small at lower temperatures and large at higher temperatures. This is a natural result. Because of the present work, the classical distributions used in the multisource thermal model [1–4] can be replaced by the revised distributions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10975095, the National Fundamental Fund of Personnel Training under Grant no. J1103210, the Open Research Subject of the Chinese Academy of Sciences Large-Scale Scientific Facility under Grant no. 2060205, and the Shanxi Scholarship Council of China.