When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
Without accounting for this distinction in GRT, several related works speak only of the attenuation law'' and produce nonreciprocal transport in bounded domains @cite_25 @cite_52 @cite_64 @cite_114 . However, the methods in this paper also apply for these formalisms.
@cite_114 @cite_52 @cite_64 @cite_25
2009182597 2114927595 2962962068 2119937051
Abstract This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ(1 R n−1). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume2(B), whereas the same traffic load scales as volume1+1 n (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic ... We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than O(n), where n denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequency-selective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of spl Omega (n) even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of independent and identically distributed (i.i.d.) channels; constant, flat channels; and constant, frequency-selective channels. The transport capacity is achieved by nodes communicating only with neighbors, and using only point-to-point coding. The thrust of these results is that the multihop strategy, toward which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open. We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance τ(s), for fixed physical distance s, thus becomes (1 + τ(s) a)− a, with standard exponential decay recovered when a → ∞. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized transport model are stated in integral form for d = 1, 2, 3 spatial dimensions. A deterministic numerical solution is developed in d = 1 using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when a = ∞. Differential diffusion equations are not presently known when a < ∞, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in d = 2, a... Wireless networks with a minimum inter-node separation distance are studied where the signal attenuation grows in magnitude as 1 ρδ with distance ρ. Two performance measures of wireless networks are analyzed. The transport capacity is the supremum of the total distance-rate products that can be supported by the network. The energy cost of information transport is the infimum of the ratio of the transmission energies used by all the nodes to the number of bit-meters of information thereby transported.If the phases of the attenuations between node pairs are uniformly and independently distributed, it is shown that the expected transport capacity is upper-bounded by a multiple of the total of the transmission powers of all the nodes, whenever δ > 2 for two-dimensional networks or δ > 5 4 for one-dimensional networks, even if all the nodes have full knowledge of all the phases, i.e., full channel state information. If all nodes have an individual power constraint, the expected transport capacity grows at most linearly in the number of nodes due to the linear growth of the total power. This establishes the best case order of expected transport capacity for these ranges of path-loss exponents since linear scaling is also feasible.If the phases of the attenuations are arbitrary, it is shown that the transport capacity is upper-bounded by a multiple of the total transmission power whenever δ > 5 2 for two-dimensional networks or δ > 3 2 for one-dimensional networks, even if all the nodes have full channel state information. This shows that there is indeed a positive energy cost which is no less than the reciprocal of the above multiplicative constant. It narrows the transition regime where the behavior is still open, since it is known that when δ < 3 2 for two-dimensional networks, or δ < 1 for one-dimensional networks, the transport capacity cannot generally be bounded by any multiple of the total transmit power.