By Frank Spitzer (auth.), Rick Durrett, Harry Kesten (eds.)
This choice of articles is devoted to Frank Spitzer at the celebration of his sixty fifth birthday. The articles, written through a bunch of his buddies, colleagues, former scholars and coauthors, are meant to illustrate the key impression Frank has had on likelihood conception for the final 30 years and probably can have for a few years to come back. Frank has continually beloved new phenomena, fresh formulations and stylish proofs. He has created or unfolded numerous study parts and it's not outstanding that many of us are nonetheless understanding the implications of his innovations. in terms of advent we've reprinted a few of Frank's seminal articles in order that the reader can simply see for himself the purpose of foundation for far of the study awarded the following. those articles of Frank's take care of houses of Brownian movement, fluctuation idea and power thought for random walks, and, in fact, interacting particle platforms. The final sector used to be began via Frank as a part of the overall resurgence of treating difficulties of statistical mechanics with rigorous probabilistic tools.
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Extra resources for Random Walks, Brownian Motion, and Interacting Particle Systems: A Festschrift in Honor of Frank Spitzer
The terms inside the parentheses represent the driving forces due to gravity (qg) and the gradient of Plateau border suction B/Bz(r/rp). , when the liquid fraction (and hence rp) is smaller at the top). Leonard and Lemlich  were the ﬁrst to consider the eﬀects of surface viscosity on drainage in a Plateau border channel. However, Eq. (36), which diﬀers from their equation only by a constant factor, has been used more frequently because Desai and Kumar have provided a very simple equation for cv.
Between z = 0 and z = z1), a mass balance for the bubbling gas can be written as: Z d GA ¼ dt z2 Að1 À eÞdz þ A z1 dz1 dt ð125Þ In Eq.
Each is discussed separately. (a) Case 1: Separation of the Continuous Phase. Let us deﬁne a critical volume: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 1 À eb KUgR0 L0 1 À eb À1 À1 Vc u tan þ ð107Þ À tan eb r eb KUgR0 A comparison with Eq. (98) indicates that Vc is the volume of the continuous phase per unit cross-section in an equilibrated foam when z1e = 0 and z2e = L0 (see Fig. 10). Thus, if e0L0 = Vc, the continuous phase will redistribute itself until an equilibrium is established with eAz¼z2 ¼ eb.