Field-Theoretical Representation of Interactions between Particles: Classical Relativistic Probability-Free Kinetic Theory
Round 1
Reviewer 1 Report
The reserach is of high scientific interest for the microscopic kinetic theory of classical many-body systems. In order to connect and further clarify the relation of their results to the thermodynamical limit the authors may add a brief discussion on whether the approach allows to extract quantities like temperatur, dissipation, distribution functions etc. and how that could be done.
Author Response
Thank you very much!
Reviewer 2 Report
Report on “Field-theoretical representation of interactions between particles: classical relativistic probability-free kinetic theory”
In this work the authors consider the atomic dynamics in a (special) relativistic description which could be compatible with a static interatomic potential at rest.
Given an interatomic potential, its Fourier decomposition is modelled in terms of a spectral function, focusing for their work on a description in terms of a finite number of poles in the complex momentum square plane. This case can then be mapped into a finite number of effective Yukawa potentials. The first hypothesis adopted by the authors is to define a dynamics introducing a scalar field for each Yukawa interaction and consider the relativistic coupled system of atoms (particles) and scalar fields.
A specific model is proposed with scalar fields with different masses sourced only by the particles with specific couplings, in the form given in eq. (28), considering retarded green functions.
Equations for the distribution in phase space of the particles are derived.
The work is an extension of previous works by the authors with the novelty of dealing with relativistic regimes.
I have the following remarks.
1) There is one important point which seems not clear in this work.
Starting from their defining model in eq. (28) the authors derive the equation of motion for each scalar field as in (31). It is stated that the right hand side of such equation, written as (33), is independent of the type of scalar field s. But this is not true in general. It is true only if one assumes that the ratios \gamma_s \over \chi_s are independent of s, as in the case of eq. (34). These conditions are a very strong assumption added ton top of eq. (28), and not a consequence of it. This should be clearly stated with a discussion about the physical origin of such conditions.
2) Given the emphasis put by the authors on the novelty and new features of their relativistic approach compared to the non relativistic one, it would be very important to discuss at least the non relativistic limit of their equations together with the first corrections beyond the classical non relativistic description.
3) Finally the model presented by the authors can be interpreted as an effective dynamical description constructed from static informations. It should be important to have support for such an effective dynamical description from extra informations, i.e. some dynamical observables, for some simple specific physical system, both in the non relativistic and relativistic regimes. The authors should discuss such kind of constraints, which, if found to be satisfied in some cases, could be of strong support to their approach.
The authors are asked to revise their paper to discuss the points raised above.
I think that, provided this is done in sufficient details, the work could be suitable for publication.
Author Response
1) There is one important point which seems not clear in this work.
Point 1: Starting from their defining model in eq. (28) the authors derive the equation of motion for each scalar field as in (31). It is stated that the right hand side of such equation, written as (33), is independent of the type of scalar field s. But this is not true in general. It is true only if one assumes that the ratios \gamma_s \over \chi_s are independent of s, as in the case of eq. (34). These conditions are a very strong assumption added ton top of eq. (28), and not a consequence of it. This should be clearly stated with a discussion about the physical origin of such conditions.
Response 1: Thank you very much! This remark is absolutely right! Our assumption presented in formula (33) of the original version of the manuscript does not hold in general case. We performed a thorough review of our manuscript and made significant corrections to it. The corrected part is marked with a vertical line in the right margin of the new version of the manuscript (formulas (31-34) and the text between them, as well as formulas (41-41) and the text between them).
Point 2: Given the emphasis put by the authors on the novelty and new features of their relativistic approach compared to the non relativistic one, it would be very important to discuss at least the non relativistic limit of their equations together with the first corrections beyond the classical non relativistic description. Considering the emphasis placed by the authors on the novelty and new features of their relativistic approach compared to the nonrelativistic one, it would be very important to discuss at least the nonrelativistic limit of their equations together with the first corrections outside the classical nonrelativistic description.
Response 2: Thanks for the interesting remark! The nonrelativistic limit of equations with corrections in lower orders is, of course, interesting to consider. However, the purpose of our work is not limited to the construction of a relativistic kinetic theory. The main points of our work are as follows:
(i). Relativism leads to the need for the field concept of interactions between particles and the idea of the field as an integral part of the system.
(ii). Possible exclusion of the concept of probability as a source of thermodynamic behavior of a system of interacting particles and the corresponding microscopic substantiation of the laws of thermodynamics. In our previous works, it was shown that the relativistic effect of interaction delay contains the mechanism for establishing thermodynamic equilibrium and the phenomenon of irreversibility.
(iii). The study of dynamics not in terms of the usual statistical distribution functions, but in terms of exact microscopic distribution functions and subsequent analysis of the qualitative properties of the solutions of the resulting equations. Microscopic distribution functions of a system of particles were introduced earlier in very few works (Kadomtsev, B.B. Fluctuations in Gases. J. Exptl. Theoret. Phys. U.S.S.R. {1957}, {5}, 771--772.), Yu.L. Klimontovich [26] and others, but almost always there was a transition to the probabilistic concept.
Therefore, it seems interesting to fully analyze the relationships between our approach and other options, which is hardly possible within the framework of one short article.
Point 3: Finally the model presented by the authors can be interpreted as an effective dynamical description constructed from static informations. It should be important to have support for such an effective dynamical description from extra informations, i.e. some dynamical observables, for some simple specific physical system, both in the non relativistic and relativistic regimes. The authors should discuss such kind of constraints, which, if found to be satisfied in some cases, could be of strong support to their approach.
Response 3: Thanks for the interesting remark! Of course, research in this direction, including your remark, will be continued, and we will present the results for publication in future articles.
Round 2
Reviewer 2 Report
I appreciate the response of the authors to my first report.
Since they agree with my comment in points 2 and 3 even if they do not want to add investigation in this work along those directions, they could at least mention them as future planned line of research.
There is a further point which requires some clarifications for the readers, which regards the relativistic distribution function in eq. (36) which satisfies eq. (37). Indeed in the latter there is an explicit dependence on the mass m, which was never introduced. The natural implicit hypoythesis is that all the "particles" with mass $m_a$ are of the same nature and of equal mass. In any case the authors should make a statement.
After those revisions are implemented the paper could be published in the Journal.
Author Response
Point 1: Since they agree with my comment in points 2 and 3 even if they do not want to add investigation in this work along those directions, they could at least mention them as future planned line of research.
Response 1: Thanks for the useful remark! At the end of the “Discussion and conclusion” section, we listed the planned directions for our future research in this area.
Point 2: There is a further point which requires some clarifications for the readers, which regards the relativistic distribution function in eq. (36) which satisfies eq. (37). Indeed in the latter there is an explicit dependence on the mass m, which was never introduced. The natural implicit hypoythesis is that all the "particles" with mass $m_a$ are of the same nature and of equal mass. In any case the authors should make a statement.
Response 2: Thank you for this remark! Indeed, this is our unfortunate omission. To remedy this, we have made two amendments.
- In the text after formula (34) we have replaced the words "system of particles" with "system of identical particles".
- In the text between formulas (37) and (38) we added the words "$m$ is the mass of each of the particles"
Author Response File: 
Author Response.pdf
