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Comptes Rendus Mathématique

Binary quantum collision operators conserving mass momentum and energy

Pierre Degond, Christian A. Ringhofer

Published Apr 30, 2003• 11 citations
Publisher

Abstract

In this Note, we generalize the Boltzmann collision operator modeling binary particle–particle collisions to a quantum framework using nonlocal quantum entropy principles.
DOI
10.1016/s1631-073x(03)00185-7

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We have argued in previous notes that C(T) exp(-ei/T) (the Maxwell-Boltzmann probability) and exp(-iEt) exp(ipx) (the free particle quantum probability called a wavefunction) have the same informational structure in that product probabilities for a given E1=Ei+Ej for the MB case and Ei,pi and Ej, pj for the quantum case lead to uniform distributions, i.e. ones with minimal information if energy and momentum are conserved. We suggest here that the root for this may be same. We have argued in previous notes that dp/dt = Force cannot have dt be infinitesimally small. This means that there exists a scale at which dt is not 0 or even tiny. We also suggested that the force (even though obeying action-reaction in collisions) may strike at some probabilistic time within dt. We then used special relativity to find that dt =hbar/E and dp=hbar/p. A probability which demonstrates these gradations is exp(-iEt+ipx) (and also Lorentz invariant). This is akin to the notion that p(e1)p(e2)=p(e3)p(e4) and p(p1)p(p2) = p(p3)p(p4) (one dimension for simplicity). In other words, the stochasticity introduced is one of a uniform distribution for product probabilities. One could obtain the same result by starting with this assumption and then suggesting Lorentz invariance. We suggest that this quantum uniform distribution product probability also holds for the Maxwell-Boltzmann case. There, the probability is not complex because p(ei) must be a real number linked to the number of particles with ei. Nevertheless, in an elastic collision of ei and ej, we suggest that the quantum result (and even dp/dt=F) suggest that there is stochasticity in x and t and that one cannot deterministically predict the outcome. Any outcome which conserves energy and momentum has the same likelihood. We suggest that it is the quantum mechanical assumption which is identical to the ideal gas case (which uses quantum mechanical gas molecules albeit at high temperatures). The high temperatures mean that one can only use the conservation laws and the associated stochasticity. We suggest that the stochasticity of interaction inherent in quantum mechanics carries over to collisions in the classical gas. It is not simply that one is approximating the situation in a gas by throwing away information. If one were to try to be as deterministic as possible at some scale (the quantum one) stochasticity arises and there is no full determinism in a collision. The best one can do is use conservation of momentum and sometimes conservation of energy in certain situations like high temperature. If this is the case, one may ask: Why is it possible to perform deterministic classical calculations such as particles bouncing off walls or mirrors or interacting with a potential V(x) in a completely deterministic manner. We suggest that these involve constrained interactions, i.e. the wall or V(x) is associated with a huge mass compared to the particle, unlike in a collision of two identical particles. Action-reaction in the former case is associated with the large mass and so always the same approximately 0 kinetic energy and hence no p(ei)p(ej) =p(ek)p(el), but in the latter, with a mass the same size and so the stochasticity may be seen clearly.

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Note Nov. 12, 2022 In (1) the discrete energy is e=bn b=constant and n=0,1,2,3... not n=1,2,3 .. as listed below. Ideas of entropy appeared before quantum mechanics was developed. In particular Planck introduced discrete energy levels: en where e= constant and n=1,2,3 to describe allowed energies of an oscillator representing allowed light in a blackbody as shown in (1). In such a case average energy and entropy may be calculated using formulas from Boltzmann. In (1) it is shown that if continuous energy is used entropy is undefined as T=temperature→0 while entropy→0 as T→0 if there are discrete energy levels. As noted this problem applies to light and uses an oscillator model, but leads to the notion of discrete energy levels. In this note we suggest using a particle in a box instead and consider only two levels for simplicity. We still use Boltzmann’s probability factor exp(- energy(i)/T) where i=1,2. In such a case the two probabilities are 1-exp(-e(i)/T) and exp(-e(i)/T). If there is a gap between e(1) and e(2) then as T→0 the particle is in state e(1) i.e. has an entropy of 0. If there is no gap then one has a problem as e(2)-e(1)-->0, but T→0 as well. Thus it seems that a gap must be introduced in order that entropy behave properly as already argued in (1). This gap for the particle in a box, however, violates Newtonian mechanics which allows all momentum p values. As a result given that a gap is needed it may also be the case that e1 does not correspond to p1=0 as in Newtonian mechanics. We used the term box above and argue that the concept of equilibrium and T seem to apply to a confined system i.e one with a length constraint for a 1D example.. For example, the Boltzmann factor exp(-e/T) may be derived considering probability densities n(e(i)) with e1+e2=e3+e4 and n(e1)n(e2) = n(e3)n(e4). This is the reaction balance approach, but it seems that the particle must be confined for such an approach to even be defined. Thus we argue that there is something “special” occurring due to this constraint i.e. confinement in space. If energy has gaps then for a particle in a box: energy = pp/2m so p=momentum itself must have gaps. Furthermore these gaps must be somehow linked to the length of the box. Thus momentum is being linked (and energy too) with length. In fact ,given that one has gaps and uniformity throughout the 1D box, one may consider the length being broken into integer equal sized portions L/n. This is already a crude notion of a wavelength and leads to the idea of allowed p momentum values being associated with L/n in some manner. Finally we note that Fermat’s principle already links an invariant space variable with momentum conservation in the same direction. The classical analogue is Maupertuis’ principle for particles with rest mass. In both cases one may link these approaches with the Lorentz invariant A=-Et+px which happens to be both the relativistic and nonrelativistic classical actions of a free particle. dA/dx partial = p so an eigenvalue equation is: -id/dx exp(ipx) = p exp(ipx) with b/p representing the wavelength. Thus one has a frequency proportional to 1/E and a wavelength proportional to 1/p. Any p, however, is allowed so there is no gap. There is, however, an association of length with p. Thus if a box with a T has a length constraint one may suggest fitting integer numbers of sine wave crests into the box to create the “gap” required by entropy considerations.

•Nov 12, 2022•0 citations
In the context of an ultracold Fermi gas, we derive conservation laws for mass, energy and momentum based on a generalized nonlocal Boltzmann equation with gradient corrections in the collision integral. The corrections are expressed in terms of effective collision duration, particle displacement and changes of total momentum and energy. Their origin is in the in-medium T matrix. Using variations of the optical theorem, we show that in the collision integral the particle-hole symmetry can be recast into a form of collision symmetry amenable to semiclassical simulation. Pauli-blocked collisions are distinguished from Bose-stimulated nondissipative ones; the latter are not present in the absence of gradient corrections. Consolidating with the microscopic theory, we extract local conservation laws for a general time-dependent interaction potential, and demonstrate how both types of collisions affect densities and flows of conserving quantities. Comparison is made with the approach of Nozières and Schmitt-Rink in the limit of thermal equilibrium. Under approximations used for normal-state ultracold Fermi gases interacting via Feshbach resonances we demonstrate the effect of the collision delay on the shear viscosity.
We have argued in previous notes that C(T) exp(-ei/T) (the Maxwell-Boltzmann probability) and exp(-iEt) exp(ipx) (the free particle quantum probability called a wavefunction) have the same informational structure in that product probabilities for a given E1=Ei+Ej for the MB case and Ei,pi and Ej, pj for the quantum case lead to uniform distributions, i.e. ones with minimal information if energy and momentum are conserved. We suggest here that the root for this may be same. We have argued in previous notes that dp/dt = Force cannot have dt be infinitesimally small. This means that there exists a scale at which dt is not 0 or even tiny. We also suggested that the force (even though obeying action-reaction in collisions) may strike at some probabilistic time within dt. We then used special relativity to find that dt =hbar/E and dp=hbar/p. A probability which demonstrates these gradations is exp(-iEt+ipx) (and also Lorentz invariant). This is akin to the notion that p(e1)p(e2)=p(e3)p(e4) and p(p1)p(p2) = p(p3)p(p4) (one dimension for simplicity). In other words, the stochasticity introduced is one of a uniform distribution for product probabilities. One could obtain the same result by starting with this assumption and then suggesting Lorentz invariance. We suggest that this quantum uniform distribution product probability also holds for the Maxwell-Boltzmann case. There, the probability is not complex because p(ei) must be a real number linked to the number of particles with ei. Nevertheless, in an elastic collision of ei and ej, we suggest that the quantum result (and even dp/dt=F) suggest that there is stochasticity in x and t and that one cannot deterministically predict the outcome. Any outcome which conserves energy and momentum has the same likelihood. We suggest that it is the quantum mechanical assumption which is identical to the ideal gas case (which uses quantum mechanical gas molecules albeit at high temperatures). The high temperatures mean that one can only use the conservation laws and the associated stochasticity. We suggest that the stochasticity of interaction inherent in quantum mechanics carries over to collisions in the classical gas. It is not simply that one is approximating the situation in a gas by throwing away information. If one were to try to be as deterministic as possible at some scale (the quantum one) stochasticity arises and there is no full determinism in a collision. The best one can do is use conservation of momentum and sometimes conservation of energy in certain situations like high temperature. If this is the case, one may ask: Why is it possible to perform deterministic classical calculations such as particles bouncing off walls or mirrors or interacting with a potential V(x) in a completely deterministic manner. We suggest that these involve constrained interactions, i.e. the wall or V(x) is associated with a huge mass compared to the particle, unlike in a collision of two identical particles. Action-reaction in the former case is associated with the large mass and so always the same approximately 0 kinetic energy and hence no p(ei)p(ej) =p(ek)p(el), but in the latter, with a mass the same size and so the stochasticity may be seen clearly.
Note Nov. 12, 2022 In (1) the discrete energy is e=bn b=constant and n=0,1,2,3... not n=1,2,3 .. as listed below. Ideas of entropy appeared before quantum mechanics was developed. In particular Planck introduced discrete energy levels: en where e= constant and n=1,2,3 to describe allowed energies of an oscillator representing allowed light in a blackbody as shown in (1). In such a case average energy and entropy may be calculated using formulas from Boltzmann. In (1) it is shown that if continuous energy is used entropy is undefined as T=temperature→0 while entropy→0 as T→0 if there are discrete energy levels. As noted this problem applies to light and uses an oscillator model, but leads to the notion of discrete energy levels. In this note we suggest using a particle in a box instead and consider only two levels for simplicity. We still use Boltzmann’s probability factor exp(- energy(i)/T) where i=1,2. In such a case the two probabilities are 1-exp(-e(i)/T) and exp(-e(i)/T). If there is a gap between e(1) and e(2) then as T→0 the particle is in state e(1) i.e. has an entropy of 0. If there is no gap then one has a problem as e(2)-e(1)-->0, but T→0 as well. Thus it seems that a gap must be introduced in order that entropy behave properly as already argued in (1). This gap for the particle in a box, however, violates Newtonian mechanics which allows all momentum p values. As a result given that a gap is needed it may also be the case that e1 does not correspond to p1=0 as in Newtonian mechanics. We used the term box above and argue that the concept of equilibrium and T seem to apply to a confined system i.e one with a length constraint for a 1D example.. For example, the Boltzmann factor exp(-e/T) may be derived considering probability densities n(e(i)) with e1+e2=e3+e4 and n(e1)n(e2) = n(e3)n(e4). This is the reaction balance approach, but it seems that the particle must be confined for such an approach to even be defined. Thus we argue that there is something “special” occurring due to this constraint i.e. confinement in space. If energy has gaps then for a particle in a box: energy = pp/2m so p=momentum itself must have gaps. Furthermore these gaps must be somehow linked to the length of the box. Thus momentum is being linked (and energy too) with length. In fact ,given that one has gaps and uniformity throughout the 1D box, one may consider the length being broken into integer equal sized portions L/n. This is already a crude notion of a wavelength and leads to the idea of allowed p momentum values being associated with L/n in some manner. Finally we note that Fermat’s principle already links an invariant space variable with momentum conservation in the same direction. The classical analogue is Maupertuis’ principle for particles with rest mass. In both cases one may link these approaches with the Lorentz invariant A=-Et+px which happens to be both the relativistic and nonrelativistic classical actions of a free particle. dA/dx partial = p so an eigenvalue equation is: -id/dx exp(ipx) = p exp(ipx) with b/p representing the wavelength. Thus one has a frequency proportional to 1/E and a wavelength proportional to 1/p. Any p, however, is allowed so there is no gap. There is, however, an association of length with p. Thus if a box with a T has a length constraint one may suggest fitting integer numbers of sine wave crests into the box to create the “gap” required by entropy considerations.