Accelerating solutions to the diffusion equation
AAccelerating solutions to the diffusion equation
Felipe A. Asenjo ∗ and Sergio A. Hojman
2, 3, 4, † Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago 7491169, Chile. Departamento de Ciencias, Facultad de Artes Liberales,Universidad Adolfo Ib´a˜nez, Santiago 7491169, Chile. Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile. Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile.
We report accelerating diffusive solutions to the one–dimensional diffusion equation with a con-stant diffusion coefficient. The maximum values of the density evolve in an accelerating fashion. Wealso construct a Gaussian–modulated form of the solution that retains this feature.
PACS numbers:Keywords:
Diffusion is one of the most important problems inphysics. Its applications goes from random walk and heat[1], economy [2] to biophysics [3]. For a one–dimensionalsystem with a diffusing material with density φ ( t, x ), andconstant diffusive coefficient D , its dynamics is given bythe equation ∂φ∂t = D ∂ φ∂x , (1)which is equivalent to the heat equation (where φ repre-sents the temperature).The standard well–known solution of Eq. (1) has theGaussian form φ ( t, x ) = (1 / √ πt ) exp (cid:0) − x / Dt (cid:1) . Thiswork is devoted to present a different exact solution ofEq. (1) with accelerating properties in the t − x plane.Several solutions for the nonlinear version of the diffusionequations have been studied (see for example Refs. [4–12]among others), and also in quantum systems [13]. How-ever, the simplest linear model described by Eq. (1) hasa solution that has not been fully explored as yet. Thissolution corresponds to an initial value condition for thetemperature φ (0 , x ) that exhibits an accelerating form ofdiffusion. It may be obtained by a the Wick rotation intime of the Berry–Balazs’s nondiffracting wavepacket so-lution [14] for non–relativistic quantum mechanics. It isa straightforward matter to show that Eq. (1) is solvedby φ ( t, x ) = Ai (cid:0) k x + k D t (cid:1) exp (cid:18) k D t x + 23 k D t (cid:19) , (2)where Ai is the Airy function, and k is an arbitrary con-stant with units of inverse length.Solution (2) presents diffusion with acceleration on x - t plane. This can be explicitly seen in Fig. 1(a) and (b). InFig. 1(a), the density (2) is displayed, in terms of dimen-sionless variables k x and k D t . It shows the oscillatingproperties of the Airy function. In the case of the heat ∗ Electronic address: [email protected] † Electronic address: [email protected]
FIG. 1: (a) Density (2), in terms of variables k x and k D t .(b) Magnitude | φ | of density (2) as it diffuses in time. Redlines represent the evolution of the position of local maximumlobes, which show accelerate in the t − x plane.. equation, these oscillations represent different tempera-ture gradients which can be set as initial conditions.The accelerating nature of the solution can be seen inFig. 1(b), where it is shown the magnitude | φ | of den-sity (2), as it diffuses in time. Red lines represent theevolution in time of the amplitudes for local maxima ofthis solution, located at different positions x M ( t ). Thecurved trajectories in each maximum represent its accel- a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug erating behavior, with d x M /dt (cid:54) = 0. In particular, themain maximum lobe diffuses with d x M /dt < φ ( t, x ) = Ai (cid:0) k x + k D t + 2 η k D t (cid:1) × exp (cid:32) k D t x + 23 k D t + ηkx + 2 ηk D t + η k D t (cid:33) , (3)is a solution to the diffusion equation (1) for an arbitrarydimensionless constant η . Solution (3) approaches a Gaussian profile as η in-creases, but retaining the accelerating features of thiskind of diffusion. This can be seen in Fig. 2(a), wherethe density (3) is shown in terms of kx and k Dt , for η = 1 /
2. In Fig. 2(b), the magnitude of density (3) ispresented for η = 1 /
2. The accelerating trajectory forthe diffusion of the maximum lobe (red line) shows thatthis feature is still present. However, it is worth notingthat as η increases, the solution becomes more Gaussian–like and the acceleration is attenuated.Solutions (2) and (3) show that accelerating propertiesfor diffusion as described by Eq. (1), are possible in a lin-ear diffusive material. These novel to diffusion processespresent interesting features that deserve to be exploredin detail. Furthermore, we expect to have other possi-ble accelerating solutions for the nonlinear version of thediffusion equation. These endeavors are left for futurework. [1] J. Crank, The Mathematics of Diffusion (Oxford Univer-sity Press, 1975).[2] F. Black and M. Scholes, Jour. Political Economy ,637 (1973).[3] D. B. Chang et al. , Jour. Theo. Bio. , 285 (1975).[4] J. M. Burgers, The Nonlinear Diffusion Equation:Asymptotic Solutions and Statistical Problems (SpringerNetherlands, 1974).[5] J. R. King, J. Phys. A: Math. Gen. , 3681 (1990).[6] J. R. King, J. Phys. A: Math. Gen. , 3213 (1991).[7] F. M. Cholewinski and J. A. Reneke, Elec. Jour. Diff.Eqs., , 1 (2003).[8] R. Metzler and J. Klafter, Europhys. Lett. , 492(2000). [9] Chai Hok Eab and S. C. Lim, J. Phys. A: Math. Theor. , 145001 (2012).[10] E. A. Saied and M. M. Hussein, J. Phys. A: Math. Gen. , 2761 (1980).[12] J. M. Hill, A. J. Avagliano, M. P. Edwards, IMA Journalof Applied Mathematics , 283 (1992).[13] R. Tsekov, Phys. Scr. , 035004 (2011).[14] M. V. Berry and N. L. Balazs, Am. J. Phys. , 264(1979).[15] G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. , 979 (2007).[16] J. Lekner, Eur. J. Phys. L43 (2009).
FIG. 2: (a) Density (3), in terms of variables k x and k D t ,and for η = 1 /
2. Its Gaussian-like form is displayed. (b)Magnitude of density (3) for η = 1 //