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R.A. Fisher , Ann. Eugenics 7 , 353 ~ 19 37 Kolmogoroff, I.Petrovsky, and N. Piscounoff,

R.A. Fisher , Ann. Eugenics 7 , 353 ~ 19 37 Kolmogoroff, I.Petrovsky, and N. Piscounoff, Moscow Univ. Bull. Math. 1 , 1 - 19 37 . P.W. Anderson, Phys. Rev. 109 , 1492 ~ 19 58 . M. Doi , J. Phys. A 9 , 1465 ~ 19 76 ; H.K. Janssen , Z. Physik. 42 , 141 ~ 19 81 .

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R.A. Fisher , Ann. Eugenics 7 , 353 ~ 19 37 Kolmogoroff, I.Petrovsky, and N. Piscounoff,

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  1. R.A. Fisher, Ann. Eugenics 7, 353 ~1937 Kolmogoroff, I.Petrovsky, and N. Piscounoff, Moscow Univ. Bull. Math. 1, 1-1937. P.W. Anderson, Phys. Rev. 109, 1492 ~1958. M. Doi, J. Phys. A 9, 1465 ~1976; H.K. Janssen, Z. Physik. 42, 141 ~1981. P. Grassberger, Z. Phys. B: Condens. Mat 47, 465 ~1982 L. Peliti, J. Phys. ~France! 46, 1469 ~1985. M. Kardar, G. Parisi, and Y.-C. Zhang, PRL 56,889 ~1986. J.L. Cardy and U.C. Tauber, PRL 77, 4780 ~1996 D.C. Mattis, M.L. Glasser, Rev. Mod. Phys. 70, 979 ~1998

  2. Q- Discreteness / microscopic fluctuations were known to • influence the approach to the equilibrium state (e.g. Fisher waves; annihilation) • Make perturbative corrections the value of a phase transition point. • Doi, Janssen, Grassberger, Peliti, Zeldovich, Michailov, Cardy, Mattis and Glasser etc etc • SO What is the novelty? • A- Here the very character of the final state is totally changed (for all values)- Discreteness makes the difference between life and death. N.M. Shnerb, Y. Louzoun, E. Bettelheim, and S. Solomon, Proc. Natl. Acad. Sci. 97, 10322 ~2000.

  3. For Experts(usually they can ask, but in such a big room I have to anticipate their thoughts): Don’t look for cheap escapes: Once a continuous a(x,t) is accepted, the death sentence for la 0 – m< 0is unavoidable Q- slow a(x,t)  a0convergence: A- it is enough a(x,t) < m/ l to have decay at all x Q- non-linear features in PDE b. = (al- m)b + Db D bA-the equation is linear in b Q -instability of the homogenousb(x,t)= b(0,t) solution: A- The solution is stable for la 0 – m< 0

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