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NANOSYSTEMS:
PHYSICS, CHEMISTRY, MATHEMATICS Original article
Matrasulov J., et al. Nanosystems: Phys. Chem. Math., 2023,14 (4), 421-427.
http://nanojournal.ifmo.ru DOI 10.17586/2220-8054-2023-14-4-421-427
Fast forward evolution in heat equation: Tunable heat transport in adiabatic regime
Jasur Matrasulov1", Jambul R. Yusupov2,6, AbdullaA. Saidov3,c
1Faculty of Physics, National University of Uzbekistan, Tashkent, Uzbekistan 2Kimyo International University in Tashkent, Tashkent, Uzbekistan 3Turin Polytechnic University in Tashkent, Tashkent, Uzbekistan
[email protected],[email protected], [email protected]
Corresponding author: Matrasulov J., [email protected]
PACS 03.65.Ta, 32.80.Qk, 37.90. j, 05.45.Yv
Abstract We consider the problem of fast forward evolution of the processes described in terms of the heat equation. The matter is considered on an adiabatically expanding time-dependent box. Attention is paid to acceleration of heat transfer processes. So called shortcuts to adiabaticity, implying fast forwarding of the adiabatic states are studied. Heat flux and temperature profiles are analyzed for standard and fast forwarded regimes.
Keywords Heat equation, shortcuts to adiabaticity, heat transport
For citation Matrasulov J., Yusupov J.R., Saidov A.A. Fast forward evolution in heat equation: Tunable heat transport in adiabatic regime. Nanosystems: Phys. Chem. Math., 2023,14 (4), 421-427.
1. Introduction
Controlling the evolution of physical processes is of fundamental and practical importance for modern science and technology. Mathematically, the problem of tunable evolution of a given physical system can be solved by developing realistic and physically acceptable models allowing manipulations in the evolution equations describing the process. Such a task attracted special attention in quantum mechanics, where an effective prescription for acceleration (deceleration) of a quantum evolution in the Schrodinger equation has been proposed earlier in [1]. The advantage of the prescription proposed in [1] is caused by the fact that it allows to speed up the time evolution of a wave function by controlling the driving potential with resultant regulation of the additional phase of the wave-function. Slightly modified version of the prescription was applied also to acceleration of the evolution for nonlinear Schrodinger equation and quantum tunneling dynamics [2]. Later, the method was considerably improved [3,4] and applied for different systems (see, e.g., Refs. [2,5-8]). Another important aspect of fast-forward protocol proposed in [1] is the flexibility for the choice of control parameter that allows one to apply it to the broad variety of physical systems described in terms of different evolution equations. In particular, the prescription was quite effective for fast forwarding of the adiabatic evolution [3,4]. In case of the adiabatic evolution, the central problem is fast forward of the adiabatic dynamics within a given short time-scale, so called "short-cuts" to adiabaticity. Simply, shortcuts to adiabaticity (STA) are the fast control protocols to drive the dynamics of system, which are fast routes to the final results of slow, adiabatic changes of the controlling parameters of a system [9,10]. In other words, a technique which allows one to control dynamics of the adiabatic processes is called "shortcuts to adiabaticity". The concept of STA was introduced first in the Ref. [11]. So far, several types of STA control protocols, such as counterdiabatic driving [12], (also known as transitionless quantum driving [13]), inverse engineering [14], local counterdiabatic driving [15] and fast-forward [3,4] protocols have been developed. Some other approaches to the STA beyond the protocol proposed in [1] are discussed in detail in Ref. [10]. In this paper, we use fast-forward approach STA from Refs. [3,4] to develop STA protocol for the heat transport processes described in terms of the heat equation. It is important to note that the protocol was found as successful application to the non-equilibrium equation of states for the quantum gas under a rapidly moving piston [7]. Also, it lead to a simple protocol to accelerate the adiabatic quantum dynamics of spin clusters [5] and adiabatic control of tunneling states [6]. The fast-forward theory is also applicable to dynamical construction of classical adiabatic invariant [16] and to classical stochastic Carnot-like heat engine [8]. Here, we will study temperature profile and heat current for fast forward system described in terms of the heat equation on a finite interval with moving boundaries. This paper is organized as follows. In the next section, we will present a brief description of the fast-forward protocol, following Ref [1]. Section 3 presents brief recall of the heat equation on a finite interval, application of fast forward protocol to the heat equation in a time-dependent box. Finally, section 4 presents some concluding remarks.
© Matrasulov J., Yusupov J.R., Saidov A.A., 2023
2. Fast forward of adiabatic dynamics described in terms of parabolic evolution equations
Here, following Ref. [1], we will briefly recall description of the fast forward protocol for the Schrodinger equation, which is a parabolic equation. We do this for 1D confined system by focusing on adiabatic dynamics. First, let us define two systems, the so-called standard system, whose evolution is to be fast accelerated and the fast forwarded system, which is just obtained as a result of acceleration of the evolution of the standard system. Our approach can be formulated as follows:
(1) A given confining potential V0 is assumed to vary in time adiabatically and to generate a stationary system which is an eigenstate of the time-independent Schrodinger equation with the instantaneous Hamiltonian. Then, both and V0 are regularized so that they should satisfy the time-dependent Schrodinger equation (TDSE);
(2) Taking the regularized system as a standard system (to be fast forwarded), we shall change the time scaling with use of the scaling factor a (t), where the mean value a of the infinitely-large time scaling factor, a(t) will be chosen to compensate the infinitesimally-small growth rate, e of the quasi-adiabatic parameter and to satisfy a x e = finite.
Thus, consider the standard dynamics with a "deformable" trapping potential, whose shape is characterized by a slowly-varying control parameter R(t) given by
R(t) = R0 + et, (1)
with the growth rate e C 1, which means that it requires a very long time T = O ^, to see the recognizable change of
R(t). The dynamics of a charged particle confined in the field of such potential can be described in terms of the following time-dependent 1D Schrodinger equation (1D TDSE):
ih d^t0 = - ¿32^0 + Vo(x, R(t))Vo, (2)
where the coupling with external electromagnetic field is assumed to be absent. The stationary bound state ¿0 satisfies the time-independent counterpart given by
£¿0 = ^¿0 = [-12dO + Vo(x, R)
¿0- (3)
2m x j
Then, using the eigenstate ¿0 = ¿0(x, R), satisfying Eq.(3), one might conceive the corresponding time-dependent system to be a product of ¿0 and a dynamical factor as
^0 = ¿0(x,R(t))e-* X E{R{t'))dt' (4)
As it stands, however, does not satisfy TDSE (2). Therefore, we introduce a regularized system
= ¿0(x,R(t))eiee(x'R(t))e-* £ E(R(t'))dt'
= ¿0eg(x, R(t))e-* /o E(R(t'))dt' (5)
together with a regularized potential
V0reg = V0(x, R(t)) + eF(x, R(t)). (6)
The unknown 6 and V will be determined self-consistently, so that should fulfill the TDSE given by
,fc s^r = _ hL
dt = - 2m' up to the order of e.
One can rewrite ¿0(x, R(t)) using the real positive amplitude ¿0(x, R(t)) and phase n(x, R(t)) as
¿0(x, R(t)) = ¿0(x, R(t))ein(x'R(t)), (8)
and see that 6 and V satisfy the following relations:
^(¿0d*6) = - m«R4 (9)
h
V h
-T = -dRn--dxn • dx6. (10)
hm
ih ^ = - hz + VT C, (7)
Integrating Eq. (9) over x, we have
m1
S2
dx6 = SR^odx', (11)
h ¿0 J
which is the main equation of the regularization procedure. The problem of singularity due to nodes of ¿0 in Eq. (11) can be overcome, so long as one is concerned with the systems with scale-invariant potentials.
x
We now accelerate the quasi-adiabatic dynamics of in Eq.(5), by applying the external electromagnetic field. To do this, we introduce the fast-forward version of as
(x,t) = ^ (x,R(A(t)))
= (x, R(A(t)))e-* /о E(R(A(t')))dt'
(12)
with
R(A(t)) = R° + eA(t), (13)
where A(t) is the future or advanced time
t
A(t) = J a(t') dt', (14)
°
and a(t) is a magnification scale factor defined by a(0) = 1, a(t) > 1 (0 < t < TFF), a(t) = 1 (t > TFF). Suppose, T to be a very long time to see a recognizable change of the adiabatic parameter R(t) in Eq.(1), and then the corresponding change of R(A(t)) is reached in the shortened or fast-forward time TFF defined by
Tff
T = J a(t)dt. (15)
°
The explicit expression for a(t) in the fast-forward range (0 < t < TFF) can be written as
2n
a(t) = a — (a — 1) cos( _ t), (16)
T /a
where a is the mean value of a(t) and is given by a = T/TFF.
more, let us assun potentials, A]](x, t) and vFi)(x, t),
Furthermore, let us assume ^FF to be the solution of the TDSE for a charged particle in the presence of gauge
*^ = Hff^F°F -
2m (— AF°) )2 + VFF+VT) ^F,
(17)
where, for simplicity, we employ the prescription of a positive unit charge (q = 1) and the unit velocity of light (c = 1). The driving electric field is given by
Eff = -^Ap - dxVF]). (18)
Substituting Eq. (12) into Eq. (17), we find 00eg to satisfy
lh dt = 2rn y i00
+ (V]0) + Voreg - E)00eg, (19)
where V0reg = Vreg(x, R(A(t))), i.e. the advanced-time variant of Eq. (6). The dynamical phase in Eq.(12) has led to the energy shift in the potential in Eq. (19).
Rewriting 00eg in terms of the amplitude 00 and phases n + as
00eg = 00(x,R(A(t)))ei'[n(x.R(A(i)))+ee(x.R(A(i)))],
and using Eq. (20) in Eq. (19), we find A]] of O(ea) and Vj^] consisting of terms of O(ea) and O((ea)2). Now, taking the limit e ^ 0 and a ^ to with ea = v being kept finite, we obtain (see Ref. [6] for details):
a]] = -ftv(t)dx0,
(20)
FF
= — v(t)dx0 • dxn
F F m
к°) = b2 b2
— 7T" (v(t))2(5x^)2 — bv(t)dR n, (21)
2m
where TFF = O r^^ = finite. In the same limiting case as above, -Ff is explicitly given by
(22)
3. Fast forwarding the heat equation based evolution
Our aim is application of fast forward protocol to the heat equation in time-varying interval, to achieve acceleration (deceleration) of the heat transfer process. Before formulating the problem, we will briefly recall the heat equation and its solution considering it on a finite interval, (0, Z). The heat equation can be written as
S = «' £ (23)
where u(x, t) is the temperature profile, k is the heat conductivity. The boundary and initial conditions can be imposed as
u(0,t) = u(Z,t)=0, (24)
and
u(x, 0) = f (x). (25)
Exact solutions of the problem given by Eqs. (1)-(3) can be obtained by factorizing variables as
u(x, t) = X(x)T(t) = Cn sin —x, (26)
Applying the principle of superposition, one obtains the general solution of Eq.(23) [17,18]
n
n=l 1
where Cn can be calculated by using initial temperature profile as follows [17,18]:
i
Cn = - j f (x) sin n—xdx.
o
In what follows we will choose Gaussian type initial temperature profile given by
1 (x-x 0 )2 f (x) = e ^
For such initial temperature profile, the coefficients, Cn are computed numerically both for standard as well as for fast-forwarded systems.
Now, consider the fast forward evolution in the heat equation. The standard system, u(0) is given in terms of the following evolution equation:
du(0) 2 dV0)
~dT = dx2 (28)
where k denotes the thermal conductivity. We consider the heat equation on a finite-interval with time-varying boundary. The time-dependent boundary conditions are imposed as
u(0)(x = 0,t) =0, u(0)(x = L(A(t)),t) =0,
We consider the case of time-dependence of the boundary given by
L(t) = L0 + -t.
Furthermore, we assume that our physical process is adiabatic, i.e. - ^ 0. Then t can be replaced by future time given as
^-^ _ n k n 4. n —
u(x,t) = ^ Cne I2 f sin —— x (27)
A(t) = J a(r)dr. This allows us to write L(t) as
0
L(t) = L0 + -A(t).
It is easy to see that wall's position for fast-forwarded system can have arbitrary time-dependence, since the magnification factor a can be any function of time. The velocity of the wall, whose value is finite, is given by
L = -a = v, - ^ 0, a ^ to.
General solution of the heat equation for adiabatically expanding box can be written as
(0)
uK '(I
(x) = 53 Cn si
= 1
n—
-x
L(A(t))
(29)
Before applying fast forward protocol of Ref. [1], we will regularize the standard system as it was done in [4]. Then the solution of the regularized standard system can be written as
and the regularized potential is given by
/ \ //->\ n n K
4TS) = Mn (x)e L(A(t))2
V (reg)(x,t) = eF(x,L(A(t))).
(30)
(31)
The dynamical phase 0 and the potential V can be obtained from Eq. (39) and (40) of Ref. [4]. In case of the box for 0 we have:
x
dx0 =--2j u2dx.
0
Now, let us consider fast-forwarding of the regularized dynamics. The heat equation for such fast-forwarded system can be written as
du(FF) ,d2 u(FF)
dt dx2
where the definition of the fast-forwarded system is given as follows [4]:
+ VFF u(FF),
(32)
2 2 2
,(FF) = „(reg)^« = e-4
Fast-forward "potential" for our case can be found in the form:
1/ da a 2 2 d0 1 2 2^a\2 VFF =--e0 — a2e^—---a2e2( V 0)2
dt dL 2 y J
Figs. 1 (a) and 1 (b) present the plots of the temperature profile for standard and fast forwarded systems, respectively.
a) b)
0.7 0.6 0.5 , 0.4 0.3 0.2 0.1 0
0.6 0.4 0.2
!
0 -0.2 -0.4
0
10 x
Fig. 1. Temperature profiles of (a) standard and (b) fast forwarded systems. The value of the heat conductance is chosen as k = 0.5, and parameter L0 = 10 for both plots, e = 0.04, a = 100
5
15
20
x
a) b)
Fig. 2. Heat flux of standard and (b) fast forwarded systems. The value of the heat conductance is chosen as k = 0.5, and parameter L0 = 10 for both plots, e = 0.04, a = 100
a)
J. Matrasulov, J. R. Yusupov, A. A. Saidov b)
Fig. 3. Heat flux of standard (a) and fast forwarded (b) systems. The value of parameters are chosen as k = 0.5, L0 = 10 and £ = 0.04, a = 100 for both plots.
Abrupt qualitative difference between the two curves can be clearly seen. In particular, widening of the profile for fast forwarded state can be observed. Plots of the heat flux for standard and fast forwarded systems, obtained using the explicit solutions given by Eq.(29) and the numerical solution of Eq.(22), are presented in Figs. 2 (a) and 2(b), respectively. Oscillations of the heat flux become more intensive for fast forwarded system. Our analysis of similar plots for much longer (than that shown in the plot) time interval showed more rapid oscillations of fast forwarded state. Figs. 3 (a) and 3 (b) present contour plots of the heat flux for standard and fast forwarded systems vs coordinate and time for the following values of parameters k = 0.5, L0 = 10, £ = 0.04 and a = 100. For standard system, the flux is localized close to the middle of the interval, between 6 and 7, while for fast forwarded system, one can observe certain blurring of the flux, although some weak localization around the points 15 and 16. Finally, one should note that in all the plots, width of the box for fast forwarded case is larger than that for the standard state. This is caused by the fact that for fast forwarded system, we use positive time-magnification factor that causes faster expansion of the interval in fast forwarded case.
4. Conclusions
In this paper, we studied fast forward problem for the heat equation by considering the latter on adiabatically expanding time-dependent box. Masuda-Nakamura fast-forward protocol [1,3] is applied for acceleration of the evolution of heat transfer process in time-dependent box with slowly moving walls. Physically important characteristics, such as temperature profile and heat flux are computed for standard and fast forwarded regimes. Considerable differences between these characteristics for standard and fast-forwarded regimes are found. In particular, the plots of the temperature profile for standard and fast forwarded states show clear quantitative and qualitative difference of the heat transfer for these two regimes. Breaking of the space-periodicity in heat flux for the fast forwarded system is also found. As the physical realization of the above model, one can consider heat transport in harmonic crystals and in graphene nanoribbon subjected to time-periodic strain. Finally, we note that the above study can be directly extended to the cases of two- and three-dimensional counterparts.
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Submitted 17 October 2022; revised 12 June 2023; accepted 25 June 2023
Information about the authors:
J. Matrasulov - Faculty of Physics, National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan; ORCID 0000-0001-5741-9003; [email protected]
J. R. Yusupov - Kimyo International University in Tashkent, 156 Usman Nasyr Str., 100121, Tashkent, Uzbekistan; ORCID 0000-0003-1758-6805; [email protected]
A.A. Saidov - Turin Polytechnic University in Tashkent, 17 Niyazov Str.,100095, Tashkent, Uzbekistan; ORCID 0000-0003-3295-4119; [email protected]
Conflict of interest: the authors declare no conflict of interest.
