By Lebossé C., Hémery C.

Manuel scolaire de mathématiques, niveau seconde C, programmes de 1965. Géométrie. Cet ouvrage fait partie de los angeles assortment Lebossé-Hémery dont les manuels furent à l’enseignement des mathématiques ce que le Bled et le Bescherelle furent à celui du français.

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**Example text**

On the other hand, the time evolution of the initial profile γ according to γ ∗ f t is somewhat slower than the equivalent time evolution obtained in the case of classical independent random walks, where the initial profile γ evolves through the heat equation’s semigroup. For any positive time, the solution of heat equation starting from γ is positive everywhere,3 but this does not happen with γ ∗ f t . Since f and γ have compact support, for any time t > 0, the function γ ∗ f t has compact support as well, hence it is not positive everywhere.

2, we define the state space of a single QRW. In Sect. 3, we explain the dynamics of a QRW. In Sect. 4 we state the hydrodynamic limit. In Sect. 5 we state and prove the local equilibrium, which in its hand implies the hydrodynamic limit. Hydrodynamic Limit of Quantum Random Walks 41 2 The State Space of the QRW We define in this section the state space of a single QRW, which, in agreement with the postulates of the Quantum Mechanics, is a Hilbert space. Its meaning is discussed below in detail.

That is, even for initial profiles with compact support, for any positive time, the solution will be non-zero everywhere. Hence, we roughly deduce that: while the QRW is faster than its classical counterpart (in the scaling aspect), a system of independent QRW’s is slower than a system of independent classical random walks (in the macroscopic diffusion aspect). The outline of the paper is the following: in Sect. 2, we define the state space of a single QRW. In Sect. 3, we explain the dynamics of a QRW.