H theorem for contact forces in granular materials

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 77, Issue 1, 2008, Pages

  Metzger, Philip T ^a  

^a KENNEDY SPACE CENTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN EQUATION; CLOSED LOOP CONTROL SYSTEMS; ENTROPY; THEOREM PROVING; TWO DIMENSIONAL;

COHESIONLESS DISKS; CONTACT FORCES; THERMODYNAMIC SYSTEM;

GRANULAR MATERIALS;

EID: 38649135705     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.77.011307     Document Type: Article

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12. Similarly, if we apply this method to an ensemble of dilute gas collision networks (in four space+time dimensions), then the phase space variables would not include the individual momentum vectors p1 and p2 for each collision, nor the outgoing momentum vectors p1′ and p2′. Instead, it would consist of the sum of the incoming momentum vectors at each collision, P, along with the colliding momentum vector angles prior to the collision, (θ1, 1) and (θ2, 2) and after the collision (θ1′, 1′) and (θ2′, 2′). The magnitudes of the incoming and outgoing momentum vectors could then be recovered by linear algebra and trigonometry. Note that this differs from the granular case only in that there is one fewer conserved component of momentum than there are dimensions in the collision network: there are Px, Py, and Pz, but no Pt corresponding to the time dimension. Despite this difference, the method works identically for both the granular and dilute gas cases.
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