Available online at www.sciencedirect.com Chaos, Solitons and Fractals 40 (2009) 803–814 www.elsevier.com/locate/chaos Transport phenomena in nanostructures and non-differentiable space–time M. Agop a a,* , Liliana Chicos b, P. Nica a Department of Physics, Technical ‘‘Gh. Asachi’’ University, Blvd. Mangeron, Iasi 700029, Romania b Faculty of Physics, ‘‘Al.I. Cuza’’ University, Blvd. Carol I, No. 11, Iasi 700506, Romania Accepted 13 August 2007 Abstract Considering that the motion of the micro-particles takes place on continuous but non-differentiable curves, in the topological dimension DT = 1, a theoretical approach of the transport mechanisms in nanostructures is established: generalized Euler’s type equations, Schrödinger’s type equation as an irrotational motion of the Euler’s fluid, Josephson type effect, and hydrodynamic model with the current expressions and conductance quantization. The correspondence with El Naschie’s e(1) space–time is given by means of some examples (the heat transfer in nanofluids, the compatibility of the acoustic regime of the phononic spectrum with the optical one, etc.).  2007 Elsevier Ltd. All rights reserved. 1. Introduction According to [1–3], the transport of charged particles in electronic devices is generally described by kinetic models such as Boltzmann-like equations or macroscopic models of hydrodynamic or diffusion type. Due to the ongoing miniaturization of these devices, reaching the nanometric scale, the reliability of these classical models becomes doubtful as quantum effects become important. Since, at an intermediate scale, collision phenomena remain significant, one of the most challenging areas of investigation in semiconductor modeling deals with the setting-up of quantum transport models which take into account scattering effects. Though many works are concerned with the numerical simulation of ballistic quantum transport models for semiconductors (see e.g. [4,5]), a quantum theory of collisions is still under development (among other works on the quantum theory of scattering, see e.g. [6,7]). Furthermore, several attempts were made to adapt existing classical macroscopic models to quantum mechanics [8,9] but, generally, the link between the so-obtained models and a microscopic quantum description of the particle transport is to a large extent phenomenological. New ideas concerning the physical basis of self-organizing phenomena observed in the so-called intelligent materials miniaturized at micrometer and nanometer sizes gained substantial interest because of the wide range of their technological purpose. Thus, miniaturization makes possible the implementation of better computer architecture but the presence of physical limits imposed by quantum and thermal fluctuation phenomena makes such systems less and less * Corresponding author. Address: Stradela Florilor No. 2, Iasi 700514, Romania. E-mail addresses: magop@phys.tuiasi.ro, m.agop@yahoo.com (M. Agop). 0960-0779/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.055 804 M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 reliable. The recognition of these ultimate limits has lead computer scientists to seek inspiration from biology. This is because living organisms operate with functional elements that are of mesoscopic scale dimensions and actually exploit collective quantum effects and thermal energy for ensuring its ‘‘living’’ state. The hope to break the barrier of miniaturization seems to lie in the knowledge of the self-organization mechanism able to explain the self-assemblage and the working regime of the simplest organisms created by nature. In solid-state physics and electronics, a large variety of different non-equilibrium phenomena accompany the spontaneous self-assemblage of spatial and spatio-temporal patterns [10]. Thus, attention has been paid to thyristor-like semiconductor structures with large active area, as these nonlinear systems with bistable properties show several spatial and spatio-temporal current density patterns. Such semiconductors could potentially be used as multi-stable elements for integrated circuits, self-organizing devices for image recognition and image processing. It is remarkable that the instability mechanism observed in such samples exhibits some features very similar to those studied in biological or chemical media. Self-organization phenomena can be revealed also in non-crystalline materials in which metastable configuration of atoms at nanometer scale gives to the material the possibility to choose one among various pathways to change free energy [10]. This occurs when, by providing some energy, the system is pushed to modify the quasi-equilibrium phenomenon ensures the reversibility of the atomic state during energy exchange with the surrounding medium. All these results requires the development of a new ‘‘scale’’ physical theories, i.e. of non-differentiable space–time type (for details see Refs. [11,12]), in which the macroscopic scale specific to the classical quantities coexist and it is compatible, simultaneously, with the microscopic ‘‘scale’’ specific to the quantum quantities. Then (i) the semi-quantum physical theories, e.g. Wigner–Boltzmann model [8], must not be imposed, but are generated as transitions between the interaction scales; (ii) the topological dimension and implicitly, the fractal one (for details see Ref. [13]) induces the transport mechanisms; (iii) the so-called anomalies, e.g. the increases of the thermal conductivity in nanostructures [14], appear as natural phenomenon in the context of material structures self-organization by means of the spontaneous symmetry breaking (for details see Refs. [8–10,15]). In the present paper, in the framework of non-differentiable space–time theory, a theoretical approach of the transport mechanisms in nanostructures is established. 2. Non-differentiable variables: the genesis of scales In the differentiable case, the usual definitions of the derivative of a given function with respect to time are equivalent, df f ðt þ DtÞ  f ðtÞ f ðtÞ  f ðt  DtÞ ¼ lim ¼ lim Dt!þ0 dt Dt!þ0 Dt Dt ð1Þ One passes from one to the other by the transformation Dt ! Dt. In other words, the differentiable nature of the space–time implies the local differential (proper) time reflection invariance. In the non-differentiable case, two functions (df+/dt) and (df/dt) are defined as explicit functions of t and of dt [11], dfþ f ðt þ Dt; DtÞ  f ðt; DtÞ ¼ lim ; Dt!þ0 dt Dt df f ðt; DtÞ  f ðt  Dt; DtÞ ¼ lim Dt!þ0 dt Dt ð2a; bÞ the (+) sign corresponding to the forward process, while () to the backward process. In other words, the non-differentiable nature of the space–time implies the breaking of the local differential (proper) time reflection invariance. Let us apply the Eqs. (2a,b) to the coordinate functions. It results [11], dX  ¼ dx þ dn ð3Þ where dx± are the differentiable (classical) variables and dn± the non-differential variables. By averaging, the Eq. (3) becomes [11], hdX  i ¼ hdx i ð4Þ hdn i ¼ 0 ð5Þ with Using (3), the speed fields result, dX  dx dn ¼ þ dt dt dt ð6Þ M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 805 We denote (dx+/dt) = v+ the ‘‘forward’’ speed and by (dx/dt) = v the ‘‘backward’’ speed. If (v+ + v)/2 may be considered as differentiable (classical) speed, the difference between them, i.e. (v+  v)/2 is the non-differentiable speed, so that we can introduce the complex speed [11]: V¼ vþ þ v vþ  v dxþ þ dx dxþ  dx i ¼ i 2 2 2 dt 2 dt In the notations dx± = d±x, (7) becomes:   dþ þ d dþ  d V¼ i x 2 dt 2 dt ð7Þ ð8Þ that allows to define the operator: d¼ dþ þ d dþ  d i 2 dt 2 dt ð9Þ While the speed-concept is classically a single concept, if space–time is non-differentiable, we must introduce two speeds instead of one, even when going back to the classical domain. Such a two-valuedness of the speed vector is a new, specific consequence of non-differentiability that has no standard counterpart (in the sense of differential physics), since it finds its origin in a breaking of the symmetry (dt ! dt). Such a symmetry was considered self-evident up to now in physics (since the differential element dt disappears when passing to the limit), so that it has not been analyzed on the same footing as the other well-known symmetries. Note that it is actually different from the time reflection symmetry T, even though infinitesimal irreversibility implies global irreversibility [11]. Now, at the level of physical description, we have no way to favor v+ rather than v. Both choices are equally qualified for the description of the laws on nature. The only solution to this problem is to consider both the forward (dt > 0) and backward (dt < 0) processes together. The number of degrees of freedom is double with respect to the classical, differentiable description (six velocity components instead of three, see (7)). Thus, two scales are obtained, the differentiable one by means of the classical variables, dxi , and the non-differentiable one through the variables, dni . According with our previous ideas, the motion of microphysical objects takes places on continuous but non-differentiable curves. The manifolds having such properties will be further called ‘‘non-differentiable space–time’’ manifolds. 3. A special consideration in the topological dimension DT = 1 Let us assume now that the curve of movement (continuous but non-differentiable) is immersed in a three-dimensional space, and X of components X i ði ¼ 1; 3Þ is the position vector of a point on the curve. Let us also consider a function f(X, t) and the following Taylor series expansion up to the first order: i i i df ¼ f ðX þ dX ; t þ dtÞ  f ðX ; dtÞ ¼   o o i dX þ dt f ðX i ; tÞ ot oX i Using the notations, dX i ¼ d X i , the forward and backward average values of this relation take the form:   of hd f i ¼ dt þ hrf  d Xi ot ð10Þ ð11Þ We make the following stipulations: the mean values of the function f and its derivatives coincide with themselves, and the differentials d±Xi and dt are independent, therefore the averages of their products coincide with the product of average. Then (11) becomes: d f ¼ of dt þ rf hd Xi ot ð12Þ so that, using (3) in the form (4), d f ¼ of dt þ rf d x ot ð13Þ In a previous work [11] a supplementary term in the Taylor expand was considered. Than, the non-zero expressions, hdni dnl i had appeared, and fractal dimension DF must be considered. In relation (13), the zero term dni appears and fractal dimension DF is not present. The consequences of these observations will be analyzed. 806 M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 If we divide by dt, relation (13) is reduced to: d f of ¼ þ v rf dt ot Now, let us calculate (df/dt). Taking into account (9) and (14), we have:    df 1 dþ f d f dþ f d f ¼ þ i  dt 2 dt dt dt dt         1 of 1 of i of of ¼ þ vþ rf þ þ v rf  þ vþ rf  þ v rf 2 ot 2 ot 2 ot ot of vþ þ v vþ  v ¼ i rf þ 2 2 ot or using (8): df of ¼ þ V  rf ; dt ot This relation also allows us to give the definition of the non-differentiable operator: d o ¼ þV r dt ot ð14Þ ð15Þ ð16Þ ð17Þ We are applying now the principle of scale covariance, and we are postulating that the passage from classical (differentiable) mechanics to the non-differentiable mechanics can be introduced by replacing the standard time derivative d/ dt by the complex operator d/dt. As a consequence, we are able to write the equation of geodesics of the non-differentiable space–time in its covariant form: dV oV ¼ þ V  rV ¼ 0 dt ot ð18Þ This means that the non-differentiability manifests only through the complex speed field V. The fractal dimension does not appear explicitly, but implicitly only by means of the V field. From here and using the operational relation V  rV ¼ rðV 2 =2Þ  V  ðr  VÞ we obtain the Euler type equation in the non-differentiable space–time  2 dV oV V  V  ðr  VÞ ¼ 0; ¼ þr 2 dt ot ð19Þ ð20Þ By applying the ($·) operator to the previous equations and denoting X ¼ r  V; ð21Þ the vortex type equation in the non-differentiable space–time is also results dX oX ¼ þ r  ðX  VÞ ¼ 0 dt ot If the ‘‘fluid’’ is irrotational, i.e. X = $ · V = 0, we can choose V of the form: V ¼ r/ with / the complex speed potential. Then, Eq. (20) becomes  2 dV oV V ¼0 ¼ þr 2 dt ot ð22Þ ð23Þ ð24Þ and more, using the relation (23) and integrating (24) we find o/ 1 ð25Þ þ ðr/Þ2 ¼ F ðtÞ ¼ const: ot 2 with F(t) a function of time only. We realize that Euler’s Eq. (24) have been reduced to a single scalar relation (25), in which the complex speed field has been replaced by a complex speed potential. Particularly, for a function / of the form / ¼ 2iD ln w ð26Þ M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 the relation (25), with the identity, ($ ln f)2 + D ln f = Df/f, takes the form   ow F ðtÞ D2 Dw þ iD þ U w¼0 ot 2 807 ð27Þ with U ¼ D2 D ln w ð28Þ where D is a constant. Up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase of w, i.e. F(t)  0, a Schrödinger type equation is obtained, D2 Dw þ iD ow  Uw ¼ 0 ot ð29Þ as an irrotational movement of fluids. w becomes simultaneously wave-function and speed potential. D defines the differential–non-differential transition, i.e. the transition from the explicit scale dependence to scale independence. In the Nottale’s model of the scale relativity theory, D has the form [11], D = k Æ c/2, with k a length scale and c the speed of light in the vacuum. This length scale is to be understood as a structure of scale space not of standard space (for example see the definition of the Compton length [11]). 4. Application: the Josephson effect in the non-differentiable space–time Let us consider the interaction between two non-differentiable structures, e.g. two nanostructures, and the corresponding interface. According with our previous model, the interface dynamics for D ¼  h=2m0 is described by the coupled equations set, ihot w1 ¼ H 1 w1 þ Cw2 ; ihot w2 ¼ H 2 w2 þ Cw1 ð30a; bÞ with w1, w2 the wave functions, H1, H2, the Hamiltonians, and C a coupling constant characterizing the interface. Exppffiffiffiffiffiffiffi liciting the wave functions in the form, w1;2 ¼ q1;2 eih1;2 , and separating in (30a,b) the real part from the imaginary one, we obtain: oq1 oq 2C pffiffiffiffiffiffiffiffiffi q q sinðh2  h1 Þ; ¼ 2¼ hrffiffiffiffiffi 1 2 ot ot rffiffiffiffiffi oh1 H 1 C q2 oh2 H 2 C q1 cosðh2  h1 Þ; cosðh2  h1 Þ ¼  ¼  ot h ot h h q1 h q2  ð31a – dÞ From here it result the current, I ¼ qðot q1  ot q2 Þ ¼ I M sinðh2  h1 Þ pffiffiffiffiffiffiffiffiffi of amplitude, I M ¼ ð4qC=hÞ q1 q2 , and phase difference Z h1  h2 ¼ ðh2  h1 Þ0 þ h1 ½H 1 ðtÞ  H 2 ðtÞdt; ðh2  h1 Þ0 ¼ const: ð32Þ ð33a; bÞ where q is the electric charge. Particularly, by applying a gravitational voltage on the interface, H1 = q Æ V, H2 =  q Æ V, and using the substitutions, h = h2  h1, h0 = (h2  h1)0, (34) and (35a) take the forms Z ð34a; bÞ I ¼ I M sin h; h ¼ h0 þ 2qh1 V ðtÞdt The relation (34a,b) reproduces a d.c. Josephson effect for null voltage, and an a.c. Josephson effect, i.e. oscillations of the current with pulsation, x = 2qV/h, when a non-null voltage is applied to the interface. In the this last case, since any time-dependent signal admits locally a Fourier discrete decomposition [16], e.g. V ðtÞ ¼ V 0 þ V 0 cosðXt þ u0 Þ, u0 = const., (34a,b) gives:      þ1 X 2qV 0 2qV 0 Jn tþd ; ð35Þ sin nX þ I ¼ IM h hX n¼1 h, the temporal average of where d = nu0 + h0 = const. and Jn is the n-order Bessel function. For X0 = nX, X0 = 2qV0/ I(t) differs from zero, i.e. it exists a continuous component of the current, I c ¼ ð1Þn I M J n ð2qV 0 = hXÞ sinðdÞ. Then, it rehX/2q, and consequently, a negative differential resistance sults peaks of the continuous current for the voltage, Vn = n 808 M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 (for details see [17,18]). Moreover, from Eq. (35) the current of the peak n varies continuously in the range ½I M J n ð2qV 0 =hXÞ; þI M J n ð2qV 0 =hXÞ at constant voltage Vn, and the phase varies in the range [p/2, +p/2]. This means that in the interface can generate or absorb the electromagnetic waves. The correspondence with e(1) space–time is achieved if the peaks of the continuous current are obtained for the voltages, Vn = m Æ h/2q, where m corresponds to the El Naschie’s limit set [12,19–26]. According with [27,28], such result corresponds to a fractional Josephson effect. 5. Hydrodynamic model in the topological dimension DT = 1 Let us consider w ¼ form (26) becomes V ¼ v þ iu pffiffiffi pffiffiffi iS qe , where q is the amplitude and S the phase of w. Then, the complex speed field (23) in the with v ¼ 2DrS; u ¼ Dr ln q Substituting (36a,b) in (24) and separating the real and imaginary parts, we obtain:  2  ov v  u2 ou ¼ 0; þr þ rðv  uÞ ¼ 0 2 ot ot ð36a; bÞ ð37a; bÞ By multiplying (37a) with m0 and introducing the ‘‘quantum’’ potential Qð1Þ ¼ m0 D2 ðr log qÞ2 m0 u2 ¼ 2 2 the momentum transport equation is obtained   ov m0 þ ðv  rÞv ¼ rðQð1Þ Þ ot ð38Þ ð39aÞ The ‘‘quantum’’ potential depends only on the imaginary part, u, of the complex speed field V. Since u arises from space–time non-differentiability, it results that the ‘‘quantum’’ potential has the same origin. In the presence of an external potential, Ue, the momentum transport equation becomes,   ov m0 ð39bÞ þ ðv  rÞv ¼ rðU e þ Qð1Þ Þ ot By integrating, (37b) with (36b) takes the form, oq þ v  rq ¼ qP ðtÞ ot ð40Þ with P(t) a new function of time only. From here, up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase of w, i.e. P(t) = 0, one obtains the equation for the probability density, oq þ v  rq ¼ 0 ot ð41aÞ From (41a) and the probability density conservation law, it results that the fluid is incompressible, rv¼0 ð41bÞ The Eqs. (39a) and (41b) define the non-differentiable space–time hydrodynamic model in the topological dimension DT = 1. The wave function of w(r, t) is invariant when its phase changes by an integer multiple of 2p. Indeed, Eq. (36a) gives: I I m0 v dr ¼ 2m0 D dS ¼ 4pnm0 D; n ¼ 0; 1; 2; . . . ð42Þ a condition of compatibility between the differentiable and the non-differentiable scales. Particularly, for D ¼  h=2m , with m* = 2m0 the mass of the Cooper type pair [28] and  hthe reduced Planck’s constant, the relation (42) takes the usual form, »p Æ dr = nh. M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 809 6. Applications: the current density and conductance quantification in the non-differentiable space–time Using the standard method from references [7,8], the current density has the form, ðlÞ j ðlÞ ¼ qDdiff rn  lðlÞ nðlÞ rðU e þ Qð1Þ þ k B T Þ ð43Þ or, substituting (38) in (43)   m0 u2 ðlÞ þ kBT j ðlÞ ¼ qDdiff rnðlÞ  lðlÞ nðlÞ r U e  2 ð44Þ In the relations (43) and (44), Ddiff is the diffusion coefficient, l is the mobility, kB is the Boltzmann constant, and the superscript l = e, p specifies the charge type (electrons and holes). Therefore, the Eq. (44) depends not only on the clasðlÞ sical quantities, i.e. qDdiff rnðlÞ , l(l)n(l)$(Ue + kBT), but also on the non-differentiable ones l(l)n(l)$[m0(u2/2)] and on the scales transition by means of the D coefficient. The result (44) is in agreement with the one from references [9–14]. Let us show that we can derive from (39a) the quantized conductance of an ideal quasi-1D liquid. We consider the ð1Þ electron liquid adiabatically connected to two reservoirs, and we denote by vL(R) and lLðRÞ ¼ QLðRÞ the speed and the chemical potential, respectively, in the left (right) reservoir, with lL  lR = eVe. Then (39a) becomes, v2L lL v2R lR þ ¼ þ 2 m0 2 m0 ð45Þ By denoting the flow speed, v = (vL + vR)/2, and the co-moving Fermi speed, vF = (vL  vR)/2, we get from (45) the relation 2m0 vvF = eVe. By definition, the current is given by I = e Æ n Æ v, so that, by using the 1D density of states and 2m0 D ¼ h, I = em0vvF/ph = e2Ve/h, which, in the linear regime, gives the quantized conductance (per spin) G0 = I/Ve = e2/h. If we assume that only a fraction a of electrons is transmitted due to the presence of a barrier in the liquid, we can argue that, in linear response, the current is an equal fraction of the current in the absence of the barrier, i.e. I = e Æ n Æ v Æ a. The conductance is thus G = a Æ e2/h in agreement with references [29]. 7. Some correspondences with El Naschie’s e(‘) space–time In the present conjecture, the correspondence with El Naschie’s e(1) space–time is given by means of some examples: (i) Thermal conoidal oscillation modes of the nanoparticle-liquid (nP/L) interface shows that this interface is ordered as a two dimensional (2D) nonlinear vortex Toda lattice [30]. Since through the relation T/T0  cn2(u) minima and maxima of the thermal field overlap with zeroes (2m + 1)K + 2inK 0 and poles 2mK + i(2n + 1)K 0 , m, n = 0, ±1, ±2, . . . of the elliptic function cn2 of complex argument u [31,32], u¼ K z; a K¼ Z z ¼ x þ iy; p=2 0 K0 b ¼ K a du 2 2 ð1  k sin uÞ 1=2 K0 ¼ ð46a–cÞ Z 0 p=2 du ð1  k 02 sin2 uÞ1=2 ; k 2 þ k 02 ¼ 1 ð47a; bÞ and a, b lattice constants along Ox and Oy directions, the real part of the complex action, S ¼ mD ln½cnðu; kÞ ð48Þ will define the potential of the vortex lattice. D defines the fractal–non-fractal transition coefficient [11]. The dynamics of the vortex lattice is given by means of the relation (48) in Figs. (1a–c) for k2 = 0.1; 0.5; 0.9. It results that in the limit cases k ! 0 and k ! 1 the ‘vortex streets’ are generated along Ox direction, and Oy direction, respectively. Only in these limits the vortex lattice gets coherence properties (the heat transport in nP/L interface works). Let us develop this idea. The ‘‘thermal impulse’’ is P¼ dS dz ð49Þ or explicitly P ¼ mD   KðkÞ snðu; kÞdnðu; kÞ 2a cnðu; kÞ ð50Þ 810 M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 Fig. 1a–c. The dynamics of the lattice. where sn and dn are the elliptic functions [32]. Since only real quantities have direct physical meaning, in what follows we consider only the real part of the expression (50). Using now the relations of transformation for the elliptic function of complex argument into elliptic functions of real argument and introducing the notations [32], s ¼ snða; kÞ; s1 ¼ snðb; k 0 Þ; K K a ¼ x; b ¼ y a a c ¼ cnða; kÞ; c1 ¼ snðb; k 0 Þ; d ¼ dnða; kÞ; d1 ¼ dnðb; k 0 Þ; ð51Þ relation (50) becomes, P ¼ mD   KðkÞ snðu; kÞdnðu; kÞ KðkÞ scd½c21 ðd 21 þ k 2 c2 s21 Þ  s21 d 21 ðd 2 c21  k 2 s2 Þ Re ¼ mD 2a cnðu; kÞ 2a ð1  d 2 s21 Þðc2 c21 þ s2 d 2 s21 d 21 Þ ð52Þ Then, in agreement with the previous observations, the following degenerations are imposed: (i) k=1, k 0 = 0, K = 1, K 0 = p/2 for Px, i.e. Px ¼ pmD sinh a cosh a 2b cosh2 a   sin2 b ð53Þ with a¼ px ; 2b b¼ py 2a ð54Þ (ii) k = 0, k 0 = 1, K = p/2, K 0 = 1 for Py, i.e. Py ¼ pmD sin c cos c 2a cos2 c cosh2 d þ sin2 c sinh2 d ð55Þ with c¼ px ; 2a d¼ py 2b The ‘‘thermal vortex field’’ X will have, through X = m1$ · P, the non-zero component " # p2 D 1 2 cos 2cðcos2 c cosh2 d þ sin2 c sinh2 dÞ þ sin2 2c 1 sinh 2a sin 2b  Xz ¼ 4a a b ðcosh2 a   sin2 bÞ2 ðcos2 c cosh2 d þ sin2 c sinh2 dÞ2 ð56Þ ð57Þ Averaging, Rb Ra Xz dS hXz i ¼ 0R b 0R a ; dS 0 0 ð58Þ M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 811 relation (57) becomes ( " cosh pD b 1 hXz i ¼  ln 2ab a p 2 cos2 pa b pb 2a pb a cosh2 pa 2b þ cos #) ð59Þ In Fig. 2 one can see the dependence of the mean ‘‘thermal vortex field’’ on the lattice lengths a, b. This means the nP/L interface behaves like a nonlinear vortex lattice, with acoustic and optical component respectively (the first and the second term of the relation (59)) of the phononic spectrum. The field diverges for a = b, specifying an intrinsic anisotropy of nP/L interface. The existence of this anisotropy determines a anisotropy of the heat transport in nanofluids – see the experimental papers [33,34]. For a b (59) takes the approximate form hXz i pD 2a2 ð60Þ Then the acoustic component of the phononic spectrum is missing – see experimental paper [33]. The ‘‘thermal vortex flux’’ is obtained by multiplying ((59)) with S = ab. From here results ( " #) þ cos pb cos h pa U b 1 b a ¼  ln U0 a p cosh2 pa 2 cos2 pb 2a 2b ð61Þ This ratio diverges for sequences a/b = 1/2n + 1 specifying the fractal-cantorian structure of the space–time. Then, in nP/ L interface ‘‘thermal anyons’’ appear (excitations of the thermal field which satisfies the fractional statistics [35]). Therefore, the fractal-cantorian structure of the space–time implies, by means of ‘‘thermal anyons’’, the increases of the heat transport in nanofluid – see the experimental paper [34]. (ii) The acoustic and the optical regimes of the phononic spectrum are described by the Jacobi elliptic function cn2. According to the equivalence theorem [32], i.e. for the compatibility of the acoustic regime of the phononic spectrum with the optical one, between two sets of fundamental elliptic periods (x1, x2) and ðx01 ; x02 Þ, the Möbius transformation exists, l¼ al0 þ b ; cl0 þ d where a; b; c; d 2 N ; l¼ x2 ; x1 x02 x01 and ad  bc ¼ 1 ð62a–cÞ with l0 ¼ Fig. 2. The dependence of the mean ‘‘thermal vortex field’’ on the lattice lengths a and b. ð63a–bÞ 812 M. Agop et al. / Chaos, Solitons and Fractals 40 (2009) 803–814 x1 ¼ 4 Z x01 ¼ 4 Z p=2 du1 ð1  0 p=2 0 k 21 sin u1 Þ du2 ð1  k 22 sin u2 Þ x2 ¼ 2i Z p=2 ; 1=2 x02 ¼ 2i Z p=2 ; 1=2 0 du1 1=2 2 ð1  k 02 1 sin u1 Þ 0 du2 1=2 2 ð1  k 02 1 sin u2 Þ ; k 21 þ k 02 1 ¼ 1; ð64a–cÞ ; k 22 þ k 02 2 ¼ 1; ð65a–cÞ The compatibility condition of the two regimes implies c = 1 and d = 0 respectively (this fact can be directly verified using (65a–c) relations and the condition x1 ! 1). In this case, let us consider a transformation element in the matrix form:   a 1 AðÞ ¼ : ð66Þ 1 0 Now, by means of the eigenvalues (for details see [15]) we obtain the irrational roots: ðnÞ ðnÞ n n ð67a; bÞ x1 ¼ ð/Þ ; x2 ¼ ð1=/Þ pffiffiffi where / ¼ 5  1 =2 is the golden mean of El Naschie’s e(1) space–time theory [13–15]. From this result, all the consequences of the e(1) space–time theory can be applied [13–15], and more recently [36–42]. 8. Conclusions The main conclusions of the present paper are as follows: (i) In the topological dimension DT = 1 of a non-differentiable space–time, a generalized Schrödinger type equation as an irrotational movement of Euler’s type fluids is obtained. Then, w simultaneously becomes wave-function and complex speed potential. (We note that Nelson in [43] defines mean forward (Dx(t)) and backward (D*x(t)) derivatives. If x(t) is differentiable, than Dx(t) = D*x(t) = dx/dt, but in general D*x(t) is not the same as Dx(t). Consequently, in general, the speed field v(t) is not differentiable). The interface between two nanostructures can generate or absorb the electromagnetic waves (Josephson effect in the non-differentiable space–time); (ii) A hydrodynamic model in a non-differentiable space–time is built. One can stress out that the ‘‘quantum’’ potential introduced in the hydrodynamic model comes from the non-differentiability of the space–time. The current density and quantized conductance for nanostructures are established. (iii) The previous formalism can be applied only on continuous but non-differentiable curve. A fractal can not be defined as non-differentiability. It is a transfinite set and can be connected or disjointed. For example, the Brownian motion is non-differentiable, but it is not a fractal. A Cantor set is a fractal but it is totally disjointed. The scale relativity theory can not be applied to non-continuous setting. By contrast, El Naschie’s e(1) [12,19–26] can mimic the continuum, although it is totally disjointed and discontinuous at high resolution. Therefore, the El Naschie’s e(1) space–time theory can be applied to non-continuous setting. 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