Semigroup

Johnson W M and Lior H (1987), Production of shiga toxin and a cytolethal distending toxin (CLDT) by semigroups of Shigella spp. FEMS Microbiology Letters, 48, 235-238. 

Prigogine thus arrived at the idea that the description of unstable dynamical systems requires an extension of dynamics. As a result, the temporal symmetry of the theory is broken. Irreversibility appears as a result of the scission of the unitary group describing the evolution into two dissipative, nonunitary semigroups. 

MSN. 167. I. Prigogine and T. Petrosky, Semigroup representation of the Vlasov equation, J. Plasma Phys. 59, 611-618. 

Figure 4. Complex plane of the variable s. The vertical axis Rei is the axis of the rates or complex frequencies. The horizontal axis Imr is the axis of real frequencies to. The resonances are the poles in the lower half-plane contributing to the forward semigroup. The antiresonances are the poles in the upper half-plane contributing to the backward semigroup. The resonances are mapped onto the antiresonances by time reversal. Complex singularities such as branch cuts are also possible but not depicted here. The spectrum contributing to the unitary group of time evolution is found on the axis Re = 0.

Figure 5. Time evolution of the statistical average (51) according to the expansion (52) of the forward semigroup valid for t > 0 and the expansion (53) of the backward semigroup valid for t < 0. See color insert.

On the other hand, the antiresonances obtained by analytic continuation toward positive values of Rei are associated with exponential decays for negative times. The corresponding expansion defines the backward semigroup ... 

The multibaker map preserves the vertical and horizontal directions, which correspond respectively to the stable and unstable directions. Accordingly, the diffusive modes of the forward semigroup are horizontally smooth but vertically singular. Both directions decouple, and it is possible to write down iterative equations for the cumulative functions of the diffusive modes, which are known as de Rham functions [ 1, 29]... 

T. G. Kurtz, Convergence of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc., 186 (1973), pp. 259-272. 

J. H. Carruth etai, The Theory of Topological Semigroups, Volume 2 (1986)... 

The trace is taken over the bath only, as in (1.15), thereby reducing the operator in Hx to an operator in Hs. This equation defines a mapping of ps(0) onto ps(t). However, this mapping is contingent on the special choice for pB(0) and cannot, therefore be utilized again to get, e.g., ps(2r) from ps(t). There is no semigroup property and no differential equation of the type... 

A. Kossakowski, Bull. Acad. Pol. Sci., Serie Math. Astr. Phys. 20, 1021 (1971) 21, 649 (1973) V. Gorini, A. Kossakowski, and E.C.G. Sudarshan, J. Mathem. Phys. 17, 821 (1976) G. Lindblad, Commun. Mathem. Phys. 40,147 (1975). A more readable account is given by R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications (Lecture Notes in Physics 286 Springer, Berlin 1987). 

R. Alicki and K. Landi, Quantum Dynamical Semigroups and Applications, Springer-Verlag, 1987. 

Vol. 1260 N.H. Pavel, Nonlinear Evolution Operators and Semigroups. VI, 285 pages. 1987. 

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of n-level systems. J. Math. Phys., 17 821, 1976. 

D. A. Lidar, Z. Bihary, and K. B. Whaley. From completely positive maps to the quantum markovian semigroup master equation. Chem. Phys., 268 35, 2001. 

Following Ref. [73] we describe first the steps before we comment on the mathematical results. Let us start with an isometric semigroup, G(f) t > 0, appropriately defined in the Hilbert space fi. If there exists a contractive semigroup SG t > 0 (defined on fi) and an invertible linear operator A, with its domain and range both dense in k, such that... 

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