Trails in Quantum Mechanics and Surroundings January 29 - February2 , 2013

1. Trails in Quantum Mechanicsand Surroundings January 29 - February2 , 2013  Laboratori INFN di Frascati (Italy) On the connection between a Korteweg fluid and the hydrodynamic form of a Logarithmic Schrödinger-like equation Giuliana Lauro Department of Industrial and Information Engineering Second University of Naples

2. Logarithmic Schrödinger equation LSE Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys. 100, 62-93 (1976) To test the linearity of Schrödinger equation, the LSE equation posses many unique properties in order to be as close as possible to the linear equation. • it guarantees the separability of noninteracting subsystems. • the stationary states can always be normalized (multiplication by a constant also yields a solution) • the Planck relation ℏω = E holds • it has the property of allowing for the separation of variables (Namely, we may construct solutions in n dimensions by taking any product of solutions in n1 and n2 dimensions (n1 + n2= n) ) Finally, it possesses simple analytic solutions in any number of dimensions. These solutions, called Gaussons, because of the Gaussian shape, move in the absence of forces as nonspreading wave packets,similar to solitons

3. GAUSSONS The logarithmic Schrödinger equation, in the absence of external forces and for positive values of the constant b, possesses simple analytic solutions in any number of dimensions n. These solutions, called Gaussons, behave as nonspreading wave packets of Gaussian shape that move uniformly, similarly to solitons : where x0 is some initial position and l2 = 1/2b,v is some given constant velocity of translation of the wave. This solution is obtained from the stationary one by means of full Galilean invariance. NOTE : the constant b measures the strength of nonlinear long range interaction: positive b means attraction, negative b indicates repulsion. Correspondingly, in the positive case we get soliton-like solutions while, negative b causes a slight spreading of wave packets faster than quantum mechanics would predict; it has the interpretation of a diffusion force. (n = 3 , ℏ=1 , m=1, a=1) a plays no significant role , we absorb it into the wave function that amounts effectively to putting a = 1.

4. SOME COMMENTS Through some experiments the upper limit on b was set at the incredibly small value of 3.3×10 –15 eV, that practically ruled out the presence of a nonlinear term in the Schrödinger wave equation. A. Shimony, Proposed neutron interferometer test of some nonlinear variants of wave mechanics, Phys. Rev. A 20, 394 (1979). C.G. Shull, D.K. Atwood, J. Arthur, and M.A. Horne, Search for a nonlinear variant of the Schrödinger equation by neutron interferometry, Phys. Rev. Lett. 44, 765 (1980). The logarithmic Schrödinger equation outlived these defeats but not as a fundamental theory. Owing to its unique properties it has been used as an exactly soluble model of nonlinear phenomena in nonlinear optics, in nuclear physics, in the study of dissipative systems, in geophysics, and even in computer science.

5. Hydrodynamic form ofLogarithmic Schrödinger equation It is well-known (Madelung 1926) that the linear Schrödinger equation can be brought into a hydrodynamic form if one writes ψ(x,t) in terms of the wave density ρ(x, t) and the phase S(x, t) as Let’s do it for the Logarithmic Schrödinger equation. By separating the real and imaginary parts, we have : Irrotational, compressible, inviscid, isothermal fluid stream velocity What kind of fluid does it represent? Which constitutive equation?

6. Brief History of Korteweg capillary fluids In order to model fluid capillarity effects, the Dutch physicist KORTEWEG formulated in 1901 a constitutive equation for the Cauchy stress that included density gradients. Specifically, KORTEWEG proposed a compressible fluid model in which the "elastic" or "equilibrium" portion of the Cauchy stress tensor T is given by In modern terminology this KORTEWEG'S form of T is a special example of an elastic material of grade N (N =3, there are all gradients of deformations less than or equal to 3). Theories of KORTEWEG'S type have been employed not only to model capillarity effects but also to model more complex spatial interaction effects like those present in liquid-vapor phase transitions under both static and dynamic conditions. KORTEWEG'S model is incompatible with conventional thermodynamics ! J.E. Dunn & J. Serrin [ On the Thermomechanics of interstitial working. Arch. Rat. Mech. Anal. 1985, 88, 95–133], by introducing the existence of a rate of supply of mechanical energy, the interstitial working, proved thatthe purely mechanical principles of linear and angular momentum balance, as well as the purely thermal Clausius-Duhem inequality, were preservedin their standard forms ( Extended Thermodynamics).

7. Compatibility with Thermodynamics In particular, D&S proved that for an isothermal elastic material of Korteweg type of grade 3, T must have the following form, in order to be thermodynamically compatible : Interstitial working : in addition to the usual working of the surface tractions and to the flow of heat, there are allowed spatial interactions of longer range to engender a rate of supply of mechanical energy across every material surface inside the continuum body. (Note that here χ = 0)

8. LSE Madelung fluid versus Korteweg fluid of grade 3I As we can write the momentum conservation equation in the the well known form: Then it can be easily checked that in the momentum equation of our Madelung fluid , the form of T is: Korteweg–type stress tensor!! Elastic material of grade 3 Surface tension coefficient Helmotz specific free energy, invariant under group rotations

9. LSE Madelung fluid versus Korteweg fluid of grade 3II From the Gausson solution of the logarithmic Schrödinger equation in absence of external forces and in the attractive case (b>0) we obtain for the stream velocity u of the Madelung fluid v any constant velocity Hence the Madelung-Korteweg fluid equations are: Whose solution is a one-parameter family of traveling waves that propagate a same 3-D Gaussian profile at constant velocity, the advective velocity v being the parameter , i.e. l2 = 1/2b

10. COMMENTS It is important to stress that, in Literature, this fluid model belongs to a class of the so called diffuse interface models used to model mixtures of fluids with phase changes, like liquid-vapor, and with phase "boundaries" of nonzero thickness, where the diffuse (capillary) interface is represented as a transition zone of rapid but smooth density variation, and consequently density gradients appear in the stress tensor. Antanovskii L. K. (1996) Microscale theory of surface tensionPhys. Rev. E 54 6285. The surface tension is intrinsically incorporated allowing for the modeling of flows associated with the spontaneous growth of tiny bubbles, as pressure drops down, as well as for their coalescence and breakdown. It is also worthwhile to observe that when b is positive, the constitutive relation for the pressure corresponds to a suitable barotropic negative pressure p = - bρ plus contributions coming from the fluctuation in density of the environment , that is: NOTES: The quantum potential inserts capillarity in the Madelung fluid The soliton solutions, with different values of velocity, could mimic the propagation of bubbles in a mixed fluid

11. LINEAR STABILITYFrom the equilibrium state of a homogeneous fluid to the composition of localized density inhomogeneities: hydrodynamical linear stability analysis Equilibrium solution ρ = ρ0 and u =u0 , constant values. Without loss of generality we can consider the fluid at rest i.e. u0 = 0. Perturbation: By standard techniques, linearizing the equations and analyzing the disturbances into normal modes, in 1-D for simplicity, we find the following dispersion relation

12. PRINCIPLE OF EXCHANGE OF STABILITY At the marginal stable mode ω =0 (k2 = 4b, b>0), bothωr ,ωi are zero, hence, thePrinciple ofExchangeof Stabilities is valid : instabilitysets in as a steady secondary flow, namely, a "cellularconvection" steady motion, where the size of the cellsare linked to critical wavenumber |kc|= 2√b = width of the Gaussian shaped soliton solution Remark 1 : The link between the characteristic size of critical nuclei and the width of the Gaussian solution shows an internal consistency of the model Remark2: for negative b, corresponding to repulsive interaction in LSE , the equilibrium is always neutrally stable, there is no fragmentation in the homogeneous fluid , the bubbles cannot form, meanwhile for positive b, corresponding to attraction in LSE, we have the possibility of a steady secondary flow- such as in the case of the convection cells that arise when a fluid is heated from below- ( cell size λc = 2π/2 √b )

13. Application in Geophysics transition from the two-phase system magma-dissolved gas in the chamber to the rising foam at conduit's base, due to rapid decompression, at the initial stage of a volcano's eruption. S.DeMartino, M.Falanga, C.Godano, G.Lauro "Logarithmic Schroedinger-like equation as a model of magma transport" Europhysics Letters, Vol.63 N.3, 2003G Lauro , “A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrodinger equation” Geophysical and Astrophysical Fluid Dynamics, Vol. 102, Issue 4 August 2008, Taylor & Francis Eds C. Godano, M. Bottiglieri, G. Lauro “Volcanic eruptions: Initial state of magma melt pulse unloading Europhysics Letters, Vol.97 N.2, 2012

14. Thank You for Your Attention Congratulations Gianfausto!! Thanks to the organizers for allowing me this “jump in the past”.