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Kinetic Theory of Granular Gases. 5 the original derivation in Ref. . This enables us to introduce a pseudo-. Liouville equation for the N-particle phase space.
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- Kinetic Theory of Granular Gases
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- Increasing temperature of cooling granular gases
Hence, we conclude that the regime of increasing temperature of a cooling granular gas may be observed for many realistic systems. Experimental investigations of cooling granular gases are not simple since to assure force-free conditions, the action of gravity has to be suppressed. Such experiments have been performed, either under true microgravity conditions aboard the space station 52 , in parabolic flights 53 , 54 , or in sounding rockets 55 , 56 , or by means of magnetic levitation By now, however, the focus of these experiments was to check experimentally the cooling law theoretically predicted by Haff and others Hence, the experiments have been designed to assure purely repulsive interaction of the particles, that is, to keep attractive forces as small as possible.
This might be the reason why the temporary increase of temperature reported in our paper was not mentioned in any of these studies. As it follows from the estimates given above, to ensure the emergence of the regime of interest, one needs to prepare a system at particular conditions.
It is worth to note that the behavior of an aggregating granular gas with increasing temperature may be interpreted as a manifestation of a negative-specific heat: The energy of the system decreases in dissipative collisions, while the granular temperature grows, that is, the smaller the energy of the gas, the larger its temperature. The negative-specific heat characterizes equilibrium systems with long-range e. Such analogy is very interesting and tempting.
Kinetic Theory of Granular Gases
Still however, the direct application of this concept to non-equilibrium systems, addressed in our study, is to be justified by further analysis. In particular, one needs to investigate the opposite process, when temperature decreases with increasing system energy. This may be possible for an extended model, which includes shattering collisions and mechanisms of energy input into the system.
Then, if the number of particles, emerging in shattering impacts, increases faster than energy of the system, thanks to the energy input, the temperature of the gas would decrease, manifesting a negative heat capacity. To apply the concept of negative heat capacity to aggregating gases, a detailed analysis of the thermodynamic additivity 47 , 48 in their cooling and heating states will also be needed.
We leave these fascinating problems for future studies. By means of kinetic theory, we show that a force-free gas of aggregative particles behaves fundamentally different than an ordinary granular gas of purely repulsive particles.
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In both cases, the kinetic energy decreases monotonously due to dissipative collisions of the particles. These regimes are characterized by either decreasing or increasing temperature. The astonishing fact that the temperature of a gas of particles, which dissipate energy increases, may be understood as follows: although the total energy of the system decreases, the total number of particles diminishes faster due to agglomeration, yielding a boost of energy per particle.
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This surprising effect is possible if the aggregation barrier increases with the size of the aggregates i. Interestingly, the increase of temperature with decaying energy corresponds to a negative-specific heat for equilibrium systems 47 , 48 ; the application of this concept to non-equilibrium systems requires however further analysis. Technically, we derived an extended set of Smoluchowski equations: equations for concentrations of different species, n k , which correspond to standard Smoluchowski equations and equations for the partial temperatures of the species, T k.
Numerical Monte Carlo simulations confirm the predictions of our theory. To derive the first set of rate Eq. Since the restitutive non-aggregating collisions preserve the number of particles, we find. For a non-aggregating granular gas, the velocity distribution function deviates from Maxwellian; the deviation may be expressed in terms of Sonine polynomials expansion 10 — 14 , These deviations are however small and, as it has been shown in refs.
Therefore, the approximation 6 is justified. The collision integral for aggregative collisions, given by Eq. The integrals in the above equation are Gaussian and hence may be straightforwardly calculated.
Computer-aided kinetic theory and granular gases - IOPscience
We perform this calculation for a particular pair, k and j. With the substitute. We also take into account that the Jacobian of transformation from v k , v j to u , w is equal to unity.
Similarly, one can find the integral in Eq. Then the remaining integration is exactly the same as for I k agg,2 , which have been already performed, therefore we find:. To derive the second set of rate equations in the system, Eq. In the right-hand side of this equation we again encounter the Gaussian integrals, as in Eq. The coefficients P ij of Eq. Using then Eqs. Generally, it is not possible to solve analytically the infinite set of Eqs.
To understand the nature of the scaling approach, we first consider a non-rigorous, qualitative analysis and then focus on its rigorous counterpart. Exploiting the basic scaling hypothesis 45 , 46 we assume that for large time and large k a scaling solution to Eqs. Hence, we seek the solution to these equations in the form.
Turn now to more rigorous analysis. As it has been shown above, in the limit of hot granular gas as well as in the cold gas limit, the kinetic kernels of Eq. Following the approach of ref.
- N. V. Brilliantov and T. Poeschel, Kinetic Theory of Granular Gases, Oxford University Press 2004.
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Hence, Eq. This implies the following condition This condition yields Eq. The scaling relations, Eqs. DSMC is a numerical technique used to directly solve the Boltzmann equation. It was first elaborated by Bird 59 for the simulation of molecular gases and later generalized and applied to the Enskog equation for dissipative granular gases, see e.
As there is an extended introductory literature on the application of DSMC to granular gases, e. For reasons which will be clear below, our DSMC algorithm does not rely on the laboratory real time but on the number of performed collisions. Here, we use the notation,.
Increasing temperature of cooling granular gases
When the particles of a dissipative gas agglomerate, obviously, their total number is not preserved. In our simulations, we start with typically 10 8 monomers and in the course of time larger and larger particles emerge, such that after some time not monomers, but much larger particles dominate. Simultaneously, the total number of particles decreases persistently.
After a relatively short period of time it will not be possible to obtain reliable data due to the poor statistics. That is, each particle is replaced by two particles of identical velocity and mass. Effectively, this operation corresponds to doubling the system volume, V , which defines a scaling variable. Consequently, the data obtained as result of the simulation have to be re-scaled before applying Eqs.
Here, N MC is the number of particles, simulated at the current time corresponding to a certain temperature. Similarly, all other variables with index MC. This expansion of the system size allows to obtain good statistical data independently of the aggregation process. The composition of a granular gas of aggregating particles changes in time.
That is, at a given time, the system consists of N 1 monomers, N 2 dimers,…, N i i -mers. The abundance of monomers can be considered as a subsystem of particles, the same for dimers and, in general, for i -mers. For the case of agglomerating particles, we use the fact that in dilute systems, subsystems of particles behave independent of one another. Obviously, since these subsystems coexist, there are many ways to determine the laboratory time from the number of collisions via Eq.
Consequently, from the simulation results we obtain many different laboratory times, which are, of course, theoretically identical, but not numerically due to the evolution of the system and the finite system size: near the beginning of the simulation, most of the particles are monomers, therefore, the statistics of the monomers delivers the most reliable results while the statistics based on mers is unreliable since the abundance of mers is yet small.
For a given particle size, i , there is a certain interval of time when the abundance of i -mers is the largest subsystem from which we obtain the most reliable data. For earlier time, the statistics is poor since most of the particles are yet smaller than i , for later times, the statistics is poor as well since most of the i -mers vanished from the system by agglomeration. Consequently, in our simulation we determine the laboratory time from the most reliable species, I , consisting of the largest number of particles at the given time.
Therefore, we compute.
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All authors analyzed the data, discussed the results, and prepared the manuscript. Electronic supplementary material.