The connection between Bohmian mechanics and many-particle quantum hydrodynamics

Bohm developed an ansatz to transform the Schr\"odinger equation into two differential equations. One of them is a continuity equation related to the conservation of the number of particles, and the other is an equation of motion similar to the Newtonian equation of motion. This ansatz for the analysis of quantum mechanics is called Bohmian mechanics (BM). These two differential equations of BM can be derived both for single-particle systems and for many-particle systems and depend on the complete set of particle coordinates of the system. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including the derivation of three differential equations: One equation for the mass balance and two differential equations for the momentum balance. In a prework [K. Renziehausen, I. Barth, Prog. Theor. Exp. Phys. 2018, 013A05 (2018)], we extended this analysis by the case that the particle ensemble consists of different sorts of particles. For such a system, a version for each of the above-mentioned three differential equations of MPQHD can be found both for each individual particle sort and for the total particle ensemble. All these differential equations of MPQHD only depend on a single position vector. The purpose of this paper is to show how the equations mentioned above, which are related to either BM or MPQHD, are connected with each other -- therefore, we prove how all the above-mentioned differential equations of MPQHD can be derived when we use the two above-mentioned differential equations of BM as a starting point. Moreover, our discussion clarifies that the differential equations of MPQHD are more suitable for an analysis of many-particle systems than the differential equations of BM because they depend only a single position vector and not on the complete set of particle coordinates of the system.


Introduction
In 1952, Bohm used preworks of Madelung [1,2] to develop his Bohmian mechanics (BM) [3,4]. In BM, the momentum balance for a non-relativistic quantum mechanical system is described by an equation of motion, which is similar to the Newtonian equation of motion -the only difference to the Newtonian equation of motion is that in the Bohmian equation of motion an additional quantity appears and is called the quantum potential. Moreover, one can derive a continuity equation for the conservation of the number of particles. Bohm focused in his works [3,4] mainly on systems with one particle or two particles. But still, he explained in [3] an ansatz for applying his mechanics on many-particle systems. In this publication, we analyse a many-particle system and call for this case the abovementioned equation being related to the conservation of the number of the particles the many-particle continuity equation of Bohmian mechanics (MPCE of BM) and the equation of motion being similar to the Newtonian equation of motion the many-particle Bohmian equation of motion (MPBEM). In addition, in 1999 Kuzmenkov and Maksimov derived for a non-relativistic many-particle quantum system the basic physics of many-particle quantum hydrodynamics (MPQHD) [5]; this analysis included the derivation of a differential equation for the mass balance and two differential equations for the momentum balance. Due to the discussions in [6] and the following analysis in this paper, we call the differential equation for the mass balance the many-particle continuity equation of quantum hydrodynamics (MPCE of QHD). In addition, we call the first of the two momentum balance equations the many-particle Ehrenfest equation of motion (MPEEM) because its derivation can be related to the Ehrenfest theorem ( [7] and [8], p. 28ff.) -and we call the second of the two mass balance equations the many-particle quantum Cauchy equation (MPQCE) because of its analogy to the Cauchy equation of motion, which is well-known in classical hydrodynamics ( [9] and [10], p. 205). As a contrast to the above-mentioned Bohmian differential equations, which depend on the complete set of all particle coordinates, the derivation of the differential equations of MPQHD includes an averaging over coordinates of all except one particle so that they only depend on the coordinates of a single position vector. Furthermore, Kuzmenkov and his coworkers developed MPQHD further to analyze spin effects [12,13] and Bose-Einstein-Condensates [14]. In addition, applications of MPQHD were discussed for electrons in graphene [15] and plasma effects [16][17][18][19][20][21]. Hereby, in [18], it is shortly mentioned how to apply MPQHD for systems where several sorts of particle are present, and in [18][19][20][21], the equations for MPQHD were discussed for the special case of two particle sorts in a plasma. Hereby, in [18][19][20], the MPQHD for electrons and a single ion sort were discussed, and in [21], these two sorts are electrons and positrons. Moreover, in [6], we developed the methods described in [5,18] further by deriving rigorously the MPQHD equations for the general case that the particle ensemble includes several sorts of particle. As a result of this discussion, we found that there is both an MPCE of QHD, an MPEEM, and an MPQCE for each individual particle sort and for the total particle ensemble, where all these differential equations only depend on one position vector, too.
In the following analysis, we again discuss a system of many particles of different sorts. Now, the purpose of this work is to take the above-mentioned two differential equations of BM as a starting point, namely the MPCE of BM and the MPBCM, and to find then these MPQHD differential equations as an ending point: We derive both for each individual particle sort and for the total particle ensemble the MPCE of QHD and the MPQCE. As intermediate results of this derivation, we will find the MPEEM both for each individual particle sort and for the total particle ensemble, too. In addition, performing this calculation, we will have to do an averaging over the coordinates of all particles except one because -as already stated -the two differential equations of BM depend on the complete set of particle coordinates, while the differential equations of MPQHD, which we want to derive, only depend on a single position vector. By working out the details of this derivation, a gap is filled because it is known how the above-mentioned differential equations of BM and the above-mentioned differential equations of MPQHD can be found using basic quantum mechanics -but to our knowledge, a derivation of these equations of MPQHD using the above-mentioned equations of BM as a starting point has not been performed yet. Our motivation to do this derivation is that the reader can learn by following the analysis given here how on one hand these differential equations of BM, and on the other hand these differential equations of MPQHD are connected with each other. In addition, the following analysis in this publication shows that the differential equations of MPQHD are more suitable than the differential equations of BM for an analysis of many-particle systems because they only depend on one position vector while the differential equations of BM depend on the complete set of particle coordinates of the system. In Sec. 2 of this paper, we recapitulate the application of BM for our system of many particles of different sorts, and we derive the MPCE of BM and the MPBEM starting from the Schrödinger equation. Then, in Sec. 3, the proof is performed how we can use the MPCE of BM and the MPBEM as a starting point to derive the MPCE of QHD and the MPQCE both for each individual particle sort and for the total particle ensemble. Finally, in Sec. 4, a summary of the contents of this work is given.
2 Basic physics of BM for a many-particle system In this section, we refer to Bohm's approach [3] to derive the many-particle continuity equation of Bohmian mechanics (MPBE of BM) and the many-particle Bohmian equation of motion (MPBEM) -besides the original publication of Bohm, we recommend to read the discussion about Bohmian mechanics for many-particle systems in [22], p. 60 ff., too. In analogy to our discussion in [6], here, we analyze a system with N S sorts of particle, where A, B stands for any number ∈ {1, 2, . . . , N S } which is related to one sort of particle. In addition, we call any Ath sort of particle also the sort of particle A or even shorter as the sort A. There are indistinguishable N (A) particles for each sort A, and each of the N (A) particles of the sort A has a mass m A . Moreover, we do not analyze spin effects in this publication (otherwise, we would have to regard that each particle of the sort A has a spin s A ). In particular, all the following analysis is valid for the three cases that all particles are bosons, or that all particles are fermions, or that the particles of some sorts are fermions and the particles of the other sorts are bosons. We designate the position vector of the ith particle of the sort A (so i ∈ 1, 2, . . . , N (A)) as q A i and name this particle the (A, i) particle. Then, the complete set of particle coordinates Q is given by and the normalized total wave function of the system is Ψ( Q, t). This wavefunction obeys the Schrödinger equation In this equation the Hamiltonian appears as: where V ( Q, t) is a scalar potential, andˆ p A i is the momentum operator of the (A, i) particle. This operator is given byˆ where ∇ A i is the nabla operator relative to the q A i coordinate. Now, Bohm applied the following ansatz to transform the Schrödinger equation (2), in which the wavefunction is factorized as where a( Q, t) and S( Q, t) are real-valued functions.
As the next step, we insert the expression (5) for Ψ( Q, t) into the Schrödinger equation (2), then we multiply the resulting formula with exp[−iS( Q, t)/ ]. After that, we separate the real and the imaginary parts of the outcome and find these two equations: where △ A i = ∇ A i · ∇ A i is the Laplace operator relative to q A i . We initially concentrate on the transformation of Eqn. (7): We multiply this formula with 2a( Q, t)/ , and after a straightforward calculation, we find: Then, we introduce the Bohmian particle density D( Q, t), the Bohmian velocity w A i ( Q, t), and the Bohmian current density J A i ( Q, t) of the (A, i) particle: At this point, we make an excursion to the topic of the naming of some quantities which appear in our prework [6] and in this work: The different names of quantities used in [6] and in this work are listed in Tab. 1. Our motivation to rename these quantities is that the Bohmian differential equations depend on the complete set of particle coordinates Q, and the QHD differential equations depend on a single coordinate q. So, in this work, we name the quantities according to Tab. 1 Bohmian quantities if they depend on the complete set of particle coordinates Q, or QHD quantities if they depend only on a single position vector q, respectively. Turning back to our derivations, using the definitions (9), (10), and (11), we can transform Eqn. (8) into the many-particle continuity equation of Bohmian mechanics (MPCE of BM), which is related to the conservation of the number of particles. It is given by [3]: Now, we define the [ A 3N (A)]-dimensional nabla operator ∇ Q and the [ A 3N (A)]dimensional Bohmian particle current density J ( Q, t): Using these definitions, the MPCE of BM can be written in a more compact form: Symbol Naming in [6] Naming in this work D( Q, t) total particle density Bohmian particle density osmotic velocity Bohmian osmotic velocity of the (A, i) particle of the (A, i) particle Bohmian velocity of the (A, i) particle of the (A, i) particle Was not used in [6] Bohmian momentum of the (A, i) particle Was not used in [6] Bohmian current density of the (A, i) particle ρ A m ( q, t) one-particle mass density QHD mass density for the particles of the sort A for the particles of the sort A ρ tot m ( q, t) total one-particle mass density QHD total mass density v A ( q, t) mean particle velocity QHD particle velocity for the particles of the sort A for the particles of the sort A v tot ( q, t) mean particle velocity QHD particle velocity for the total particle ensemble for the total particle ensemble j A m ( q, t) mass current density QHD mass current density for the particles of the sort A for the particles of the sort A j tot m ( q, t) total particle mass current density [23] QHD total mass current density Table 1: Overview over the naming of some quantities in our prework [6] and in this work. The nomenclature used here facilitates the distinction between quantities related to BM and quantities related to QHD.
Having found the MPCE of BM, we derive the MPBEM. For this task, we divide Eqn. (6) by −a( Q, t) and bring all terms on one side. Thus, we find: Then, we define the quantum potential V qu ( Q, t) as: and one can show in a straightforward calculation by inserting the definition (9) for the Bohmian particle density D( Q, t) into Eqn. (17) that this equation for the quantum potential V qu ( Q, t) can be rewritten as ( [5] and [11], p. 8): Then, we transform the formula (16) by inserting Eqn. (18) for the quantum potential V qu ( Q, t) and by inserting Eqn. (10) for the Bohmian velocity w A i ( Q, t) of the (A, i) particle. As an intermediate result, we find the many-particle Hamilton-Jacobi equation [3]: As the next step, we apply the nabla operator ∇ Q to this scalar equation and find the following [ A 3N (A)]-dimensional equation: Now, we define the Bohmian momentum p A i ( Q, t) of the (A, i) particle and the [ A 3N (A)]dimensional Bohmian momentum p( Q, t) as: Having defined the Bohmian momentum p( Q, t), we can write the first term on the left side of Eqn. (20) in the following manner: In the following calculations of this work, x, y, and z denote Cartesian vector indices or tensor indices, and sums using the Greek variables α or β as sum variables are summing up over these indices x, y, and z. Then, for the vector component of the second term on the left side of Eqn. (20), which is related to the x component of the (B, 1) particle, we find: Moreover, using the definition (10) for the Bohmian velocity w A i ( Q, t), we find that for any α ∈ {x, y, z} the following transformation is valid: As the next step, we insert Eqn. (25) into the intermediate result (24) and find: Thus, we found the result (26) for the vector component of the second term on the left side of Eqn. (20), which is related to the x component of the (B, 1) particle. Therefore, the second term on the left side of Eqn. (20) can be written in this form: Using Eqns. (23) and (27), the addition of the first and the second term of Eqn. (20) leads to the following result: As can be realized by following the discussions in [3] and [11], p. 9, this result means that the total rate of change of the Bohmian momentum d p( Q,t) dt can be decomposed into two terms, where the first term ∂ p( Q,t) ∂t is the local rate of change of the Bohmian momentum p( Q, t) for a fixed set of particle coordinates Q and the second term is related to the effect that the flow transports the fluid elements to other positions where the Bohmian momentum p( Q, t) can differ. Combining the result (28) with Eqn. (20), we find the many-particle Bohmian equation of motion (MPBEM), which is given by [3,24]: According to the interpretation of Bohm [3], this is an equation of motion for the particles of the system similar to a Newtonian equation of motion. Hereby, the (A, i) particle moves on a trajectory, where it is at the time t at the position q A i ( Q, t) and it has the Bohmian velocity w A i ( Q, t).
Now, we have found two Bohmian equations, namely the MPCE of BM (15) and the MPBEM (29), where both depend on the complete set of particle coordinates Q.

Connection between BM and MPQHD
In this chapter, we will show that the two Bohmian equations (15) and (29) can be transformed in two quantum hydrodynamic equations: The MPCE of BM (15) can be transformed into the many-particle continuity equation of quantum hydrodynamics (MPCE of QHD) and the MPBEM (29) can be transformed into the many-particle quantum Cauchy equation (MPQCE). Hereby, both the MPCE of QHD and the MPQCE only depend on a single position vector q.
Please note that we derived the MPCE of QHD and the MPQCE already in [6], but in the derivations and discussions there, the connection between quantum hydrodynamics and Bohmian mechanics was not cleared -this gap is filled in this work. Before we start with the derivations in this chapter, we point out these two details: The first detail is that in [6], the MPCE of QHD was named in a shorter manner just as the many-particle continuity equation (MPCE) because we had not to distinguish there between a Bohmian and a quantum hydrodynamic version of the continuity equation.
The second detail is that we presented in [6] two different versions of the MPCE of QHD and the MPQCE -namely, for both the MPCE of QHD and the MPQCE there is one version for an individual particle sort A and another version for the total particle ensemble. We will show that the MPCE of BM (15) can be transformed into both versions of the MPCE of QHD. Analogously, we will show that the MPBEM (29) can be transformed into both versions of the MPQCE. Now, we will show the connection between BM and MPQHD. We start to carry out this task by transforming the MPCE of BM (15) into the MPCE of QHD for the particle sort A.
Therefore, we multiply the MPCE of BM (15) by the factor N (A)m A δ( q − q A 1 ) and integrate it over the complete set of particle coordinates Q. So, we find: Then, using the fact that the QHD mass density ρ A m ( q, t) for the particles of the sort A is given by [6] we find for the left side of Eqn. (30): For the transformation of the right side of Eqn. (30), we regard that so, it is given by: Using the volume element d Q B i for all coordinates except for the coordinate q B i , we can write the volume element d Q for the complete set of particles as As the next step, we transform Eqn. (34) by splitting up the sum B,i into the summand for the special case {B = A, i = 1} and a sum over all the remaining summands. In addition, we use Eqn. (35) for these remaining summands. Then, by applying the divergence theorem, we find that the integration over the coordinate q B i for all these remaining summands can be transformed into an integral over the system boundary surface where the wave function vanishes. Thus, these remaining summands vanish, and only the summand for the special case {B = A, i = 1} remains. By considering these ideas, we find: Now, the QHD mass current density j A m ( q, t) for the particles of the sort A given by [6,25] can be inserted in Eqn. (36), and we find the following result for the right side of Eqn. (30): Finally, we can combine the results (32) and (38) for the left side and the right side of Eqn.
(30) and get as a result the many particle continuity equation of quantum hydrodynamics (MPCE of QHD) for the particle sort A [6]: Thus, we have proven that the MPCE of QHD for the particle sort A (39) can be derived from the MPCE of BM (15).
In addition, having completed this proof, it is trivial that the MPCE of QHD for the total particle ensemble can be derived from the MPCE of BM (15), too, since in [6], it was already shown that the MPCE of QHD for the total particle ensemble can be derived just by summing up the MPCE of QHD for particles of a certain sort A over all sorts of particle. This MPCE of QHD for the total particle ensemble has the form where the quantity ρ tot m ( q, t) is the QHD total mass density given by: and the quantity j tot m ( q, t) is the QHD total mass current density given by: So, we have shown that one can take the MPCE of BM (15) as a starting point to derive both the MPCE of QHD for the particle sort A (39) and the MPCE of QHD for the total particle ensemble (40).
In the following calculation, we will take the MPBEM (29) as a starting point to derive both the MPQCE for the particle sort A and the MPQCE for the total particle ensemble. We start this calculation by taking into account the three vector components of the MPBEM (29), which are related to the (A, 1) particle. For these three vector components, the following differential equation holds: Now, we multiply Eqn. (43) by N (A)D( Q, t)δ( q − q A 1 ) and integrate it over the complete set of coordinates Q. So, we find: As the next step, we introduce the force density f A ( q, t) [5,6,26] and the quantum force density f A qu ( q, t) for the particles of the sort A ( [5,27] and cf. [11], p. 326): Using Eqns. (45) and (46), we can rewrite the right side of Eqn. (44) in the following form: As an intermediate step, we introduce the momentum flow density tensor Π A ( q, t) for the sort of particle A. As we discussed already in [6], there are several versions of this tensor Π A ( q, t) because it is not this tensor itself which is physically relevant but only its tensor divergence ∇Π A ( q, t).
Moreover, we pointed out in the mentioned reference that we recommend to use the Wyatt version of this tensor, and therefore, in this work, we will only use this tensor version. The naming of this tensor version is motivated by the form of the formula (1.57) in R. E. Wyatts book [11], p. 31, and it is given by (hereby, 1 is the unit matrix, and the operation symbol ⊗ interrelates two vectors to a dyadic product) [6]: Therefore, its Cartesian tensor elements are given by: The quantity P A ( q, t), which appears in Eqns. (48) and (49), is the scalar quantum pressure for the particles of the sort A, and this quantity is given by ( [6] and cf. [11], p. 326): In addition, in Eqn. (48), the dyadic product of a vector d A i ( Q, t) for the case i = 1 appears; this vector is defined by The quantity d A i ( Q, t) is named Bohmian osmotic velocity of the (A, i) particle corresponding to the nomenclature in [6] and [11], p. 327. The Bohmian osmotic velocity d A i ( Q, t) is the quantum analog to the Bohmian particle velocity w A i ( Q, t), and in addition, this quantity is related to the shape of D( Q, t). Moreover, in [6], we discussed that one can split the momentum flow density tensor Π A ( q, t) for the particles of the sort A in a classcial part and a quantum part: Hereby, the classical part is given by and its matrix elements are And the quantum part is given by with according matrix elements Now, we will prove that the quantum force f A qu ( q, t) is equal to the negative tensor divergence of the quantum part Π A,qu ( q, t) of the momentum flow density tensor. So, we have to show that the following equation holds, where e β , β ∈ {x, y, z} are the Cartesian basis vectors: Therefore, we transform Eqn. (46) for the quantum force density f A qu ( q, t) using Eqn. (17) for the quantum potential V qu ( q, t): Now, we transform the term appearing in the first line of Eqn. (58) by regarding in a argumentation similar to the transformation of Eqn. (34) to Eqn. (36) that only the summand for {B = A, i = 1} does not vanish for this term. In addition, we use the definition (50) for the scalar pressure P A ( q, t) and find: Note that in the last two lines of the calculation above, we rewrote the gradient of the scalar quantity P A ( q, t) into the divergence of the tensor 1 P A ( q, t) . Our next step is to transform the term in curly brackets in Eqn. (58). Using the abbreviation iα and the definition (51) of the Bohmian osmotic velocity d A i ( Q, t), we find in a straightforward calculation the following result for the x component of the term in curly brackets in Eqn. (58): So, using a dyadic product d B i ⊗ d A 1 , we can write the term in curly brackets in Eqn. (58) as: As the next step, we transform Eqn. (58) using the intermediate results (59) and (61), obtaining:  ( q, t) of the momentum flow density tensor for the particle of the sort A, Eqn. (57) can be proven as: Combining Eqns. (47) and (57), we now find for the right side of Eqn. (44) the following result: As the next task, we transform the left side of Eqn. (44). Therefore, we regard Eqns. (21) and (22) as well as the three vector components of Eqn. (28), which are related to the (A, 1)-particle, and find that the left side of Eqn. (44) can be rewritten as: Now, we analyze the x component of the Eqn. (65) in detail:  (12), the third equation line can be simplified by regarding the fact that due to the divergence theorem only the summand for {B = A, i = 1} in the double sum over B and i does not vanish, and the fourth equation line is rewritten by using a vector notation instead of the sum over α. In addition, we permute the second and the third of these five equation lines and find: Above we indicated, using within a lowered brace Eqn. (54), that one finds the term appearing in the second line of the right side of Eqn. (67) to be equal to the x component of the divergence of classical part Π A,cl ( q, t) of the momentum flow density tensor for the particles of the sort A. Moreover, we merge the terms appearing in the third, fourth, and fifth line of the right side of Eqn. (67) and recognize that the sum of these terms vanishes. Thus, we get the following result for the x component of the expression on the left side of Eqn. (44): Therefore, the expression on the left side of Eqn. (44) is given by: Now, we combine the result (69) for the left side of Eqn. (44) and the result (64) for the right side of Eqn. (44) and get: By shifting the term ∇ q Π A,cl ( q, t) to the right side of Eqn. (70) and by applying Π A ( q, t) = Π A,cl ( q, t) + Π A,qu ( q, t), one gets the many-particle Ehrenfest equation of motion (MPEEM) for the particles of the sort A that we presented already in [6]: In addition, in [6], it was shown that performing some mathematical operations using the MPCE of QHD for the particles of the sort A (39), one can transform the MPEEM for the particles of the sort A (71) into the many-particle quantum Cauchy equation (MPQCE) for the particles of this sort, which is given by [28]: In the MPQCE for the particles of the sort A the QHD particle velocity v A ( q, t) for the particles of the sort A and the pressure tensor p A ( q, t) for the particles of this sort appear. The definition of these quantities was discussed already in our prework [6]. For the following discussions, there is no need to recapitulate these definitions again, thus, we skip them in this work. Taking into account that the derivation mentioned above, i. e. the derivation of the MPQCE for the particles of the sort A (72), was already worked out in detail in our previous work, we have proven that this quantum hydrodynamic equation can be derived from the MPBEM (29) as a starting point. Now, the only remaining open task of the outlined tasks at the end of Sec. 2 is the proof that the MPQCE for the total particle ensemble can be derived from the MPBEM (29) as a starting point. Therefore, we regrad as shown above that Eqn. (29) can be used as a starting point to derive the MPEEM for the particles of the sort A (71).
As the next step, we here recapitulate shortly that we have proven already in our prework [6] that one can derive the MPQCE for the total particle ensemble using the MPEEM for the particles of the sort A (71) and the MPCE of QHD for the total particle ensemble (40): Therefore, we take into account that, analogously to the definition (42) of the QHD total mass current density j tot m ( q, t), the momentum flow density tensor Π tot ( q, t) and the force density f tot ( q, t) for the total particle ensemble are given by the sum over the correspondent quantities for the particular particle sorts: Now, we can sum up the MPEEM for a certain sort of particle A (71) over all sorts of particle and using the Eqns. (42), (73), and (74), one gets the MPEEM for the total particle ensemble as a result. This equation is given by: In addition, in a calculation using the MPCE of QHD for the total particle ensemble (40), one can now transform the MPEEM for the total particle ensemble (75) into the MPQCE for the total particle ensemble. Here, we skip the details of this calculation and only state its result, the MPQCE for the total particle ensemble [6,28]: In Eqn. (76), the quantity v tot ( q, t) is the QHD particle velocity for the total particle ensemble, and p tot ( q, t) is the pressure tensor for the total particle ensemble. Like for the corresponding quantities for a single sort of particle A, we refer for the definitions of the quantities v tot ( q, t) and p tot ( q, t) to our previous work [6].
) MPQCE for the total particle ensemble A Table 2: On the left side of this table, we present the Bohmian equations, where all quantities depend on the complete set of particle coordinates Q and the time t, and on the right side the corresponding quantum hydrodynamic equations, where all quantities depend on a single postition vector q and the time t.
So, we now summarized how the MPQCE for the total particle ensemble (76) can be derived using the MPBEM (29) as a starting point.
As an additional remark to this proof, we mention that the reader might wonder why this proof was not performed by trying to find the MPQCE for the total particle ensemble (76) by summing up the MPQCE for a certain sort of particle A (72) over all sorts of particle instead of summing up the MPEEM for a certain sort of particle A (71) over all sorts of particle. The reason why we did not choose this proceeding in our analysis above is that one does not get the MPQCE for the total particle ensemble (76) just by summing up the MPQCE for a certain sort of particle A (72) over all sorts of particle because the Eqn. (72) is a non-linear differential equation where the superposition principle is not valid.
For a good overview of the equations of BM, which we used as a starting point in this work, and the equations of MPQHD, which we here found as final results, we present them again in Tab. 2.
As a final remark, we mention this analogy between classical mechanics and quantum mechanics: In a classic system containing several interacting particles, solving the coupled Newtonian equations of motion to receive the trajectories of these particles is a promising strategy for small systems only with few particles because otherwise the system of the Newtonian equations of motion cannot be solved anymore due to its big size. Thus, for many-particle systems, using classical hydrodynamics instead is a better approach because for these systems, the size of the hydrodynamical differential equations is much smaller than the size of the system of the Newtonain differential equations. In an analogue manner, solving the MPBEM (29) for a quantum system containing several interacting particles for a calculation of all the individual Bohmian particle velocities w A i is also only a promising strategy for small systems. The reason why solving the MPBEM (29) is a good ansatz for small systems only is that for many-particle systems the MPBEM (29) is difficult to solve since for these systems, the MPBEM (29) is a high-dimensional differential equation having [ N S A=1 3N (A)] dimensions. As a contrast, for many-particle systems, solving the MPEEMs (71) and (75) for a calculation of the QHD mass current densities j A m ( q, t) and j tot m ( q, t), or solving the MPQCEs (72) and (76) for a calculation of the QHD particle velocities v A ( q, t) and v tot ( q, t), are more promising approaches than solving the MPBEM (72) because the MPEEMs and MPQCEs (71), (72), (75), and (76) are always three-dimensional equations independent of the size of the system.

Summary
In the early 1950s [3,4], Bohm used an ansatz to transform the Schrödinger equation which leads to the result that the Schrödinger equation can be split in two other differential equations: The first of these equations is a continuity equation that is related to the conservation of particles, and the second of these equations is an equation of motion being very similar to the Newtonian equation of motion except for the detail that in the Bohmian equation of motion an additional quantity appears which is called the quantum potential. These two differential equations are the basis for BM, and they can be derived both for one-particle systems and for many-particle systems, where -in the general case which we analyze in this work -the different particles belong to different sorts of particle. It is remarkable that for many-particle systems, the two above-mentioned differential equations, which we named the MPCE of BM and the MPBEM, depend on the complete set of particle coordinates. On the other hand, using basic quantum mechanics, Kuzmenkov and Maksimov developed MPQHD in 1999 [5], where an averaging over the coordinates of all particles except one is made. Using this ansatz, one can derive the MPCE of QHD, which is a differential equation related to the mass conservation of the system, and two differential equations being related to the momentum conservation of the system, namely the MPEEM and the MPQCE. We picked up this analysis of Kuzmenkov and Maksimov, and in our prework [6] we discussed in detail how to extend systematically their discussion for the situation that the system contains different sorts of particle. In this case, one can derive a version of the MPCE of QHD, the MPEEM, and the MPQCE each for the individual particle sorts and for the total particle ensemble. For all these versions of the MPCE of QHD, the MPEEM and the MPQCE, it is valid that they only depend on a single position vector. In this work, we analyzed the connection between the above-mentionend differential equations of BM and MPQHD. Using an averaging over the coordinates of all particles except one, we found out how all the versions of the MPCE of QHD and the MPQCE can be derived from the MPCE of BM and the MPBEM as a starting point -and during these calculations we got all the versions of the MPEEM as intermediate results. We think that this article will help other authors to realize how BM and MPQHD are linked. In addition, it can be realized that these fields can be differentiated from each other by the context that the differential equations related to BM depend on the complete set of particle coordinates, but the differential equations related to MPQHD depend on a single position vector only. So, for many-particle systems, the differential equations related to BM are more complicated to solve than the differential equations related to MPQHD. This is an analogy to the fact that for classical many-particle systems, the Newtonian equations of motion for the trajectories of all the particles of the system are more complicated to solve than the hydrodynamic differential equations.
[28] In [6], we wrote the MPQCEs for the sort of particle A and for the total particle ensemble in an expanded form, where the total time derivative terms were written as d v A ( q,t) dt = ∂ ∂t + v A ( q, t)∇ q v A ( q, t) and d v tot ( q,t) dt = ∂ ∂t + v tot ( q, t)∇ q v tot ( q, t). This is similar to writing the MPBEM (29) in an expanded form by inserting Eqn. (28).