Method for simulating stochastic oscillation in individual-granularity long-distance expressway traffic flow using quantum harmonic oscillator
The present invention discloses a method for simulating stochastic oscillation in an individual-granularity long-distance expressway traffic volume using a quantum harmonic oscillator, which includes: firstly, describing the speed and position of a vehicle by a quantum superposition state, and constructing an energy eigenequation of the quantum harmonic oscillator to represent movement of the vehicle; secondly, constructing an n-order Hermite polynomial based on the energy eigenequation, constructing a quantum harmonic oscillator model for simulating the stochastic oscillation in the long-distance traffic flow in a mode featuring aliasing of multiple strategies, and optimizing a solution model; and finally, constructing a mapping mechanism between the probability and the traffic volume to simulate the traffic volume. The present invention definite parameter meaning, and easy solution and calculation, and is of reference significance for modeling of traffic flow in which multiple strategies and states exist for individuals and are difficult to observe.
1. A method for simulating stochastic oscillation in an individual-granularity long-distance expressway traffic flow using a quantum harmonic oscillator, comprising the following steps: (1) describing speed and position of a vehicle by a quantum superposition state, and constructing an energy eigenequation of the quantum harmonic oscillator to represent movement of the vehicle; (2) constructing an n-order Hermite polynomial based on the energy eigenequation, constructing a quantum harmonic oscillator model for simulating a stochastic oscillation in a long-distance traffic flow with multiple driving strategies, and optimizing a solution model; and (3) constructing a mapping mechanism between a probability and a traffic volume to simulate the traffic volume, wherein each individual vehicle in the long-distance expressway traffic flow is abstracted as a quantized particle, a driving strategy selected by the individual vehicle is regarded as an independent energy state, and the transition of the driving strategy is regarded as approximately the transition of an energy level,
wherein the step (3) is implemented as follows:
N
t
=
∫
0
t
(
α
t
P
t
+
β
t
)
=
α
f
(
x
)
+
β
(
8
)
wherein Nt is the traffic volume of vehicles driving off from the expressway, a and Rare normalized coefficients, at ˜ is a number of vehicles in a single time window element determined by traffic volume density in a time window [0, t], and has a strong positive correlation with the number of vehicles driving off from the station, f3t is a number of vehicles driving off from the station in the absence of oscillation, and is the average number of vehicles driving into the expressway in the time window [0, t]; f(x) is obtained by fitting the wave function based on series Nt, and then mapping coefficients a and f3 of the probability density to the total traffic volume are estimated by the least square method.
2. The method according to claim 1 , wherein the step (1) is implemented as follows:
all vehicles run at a constant speed of v, and an ideal position of any vehicle k at a time point t is S kt ; when real speed of the vehicle is greater than or less than the ideal speed ν, and recorded as |↑ and |↓ , a real position of the vehicle is ahead of or behind an ideal position S kt , and recorded as |→ and |← ; the speed of the vehicle is represented by the superposition state V t =a t |↑+b t |↓i and the position is represented by the superposition state S t =C t |→ +d t |← i at the time point t, wherein i is an imaginary unit, a t and b t represent probability amplitudes of acceleration and deceleration, respectively, c t and d t represent probability amplitudes of the position of the vehicle being ahead of or behind the ideal position, respectively, and |a t | 2 +|b t | 2 =|c t 2 +|d t | 2 =1; the movement of the vehicle is described as a quantum harmonic oscillator with an energy eigenequation as follows:
iA
d
dt
ψ
(
x
)
=
H
ψ
(
x
)
(
1
)
wherein i is an imaginary unit; A is a constant describing the distribution of individual energy levels; ψ(x) is a wave function representing the probability amplitude of an individual appearing at a specific position; H=f(V t )+g(S t ) is Hamiltonian of the system and a core characteristic of dynamic evolution of the system; f(V t ) and g(S t ) are kinetic energy and potential energy of the harmonic oscillator, respectively.
3. The method according to claim 1 , wherein the step (2) is implemented as follows:
in a real expressway traffic flow, a following n-order Hermite polynomial is constructed based on the energy eigenequation:
ψ
n
(
x
)
=
∑
0
n
w
n
(
π
2
n
n
!
)
1
2
H
n
(
x
)
e
-
x
2
2
(
3
)
wherein n is a number of energy levels representing a number of different driving strategies which can be selected by a driver in the driving process, H n (x) is an n-order Hermite polynomial, W n is a fitting parameter of the wave function representing the probability amplitude of the harmonic oscillator at different energy levels; in quantum mechanics, the probability can be expressed as square of wave function, so that the probability of the vehicle appearing at a specific position in the long-distance traffic flow can be expressed as:
P
=
❘
"\[LeftBracketingBar]"
ψ
n
(
x
)
❘
"\[RightBracketingBar]"
2
=
(
∑
0
n
w
n
(
π
2
n
n
!
)
1
2
H
n
(
x
)
e
-
x
2
2
)
2
(
4
)
equation (4) is a quantum harmonic oscillator model of the long-distance traffic flow
h
n
(
x
)
=
H
n
(
x
)
e
-
x
2
2
(
π
2
n
n
!
)
1
2
is converted to a probability expression of the Hermite polynomial, wherein h n (x) reflects the oscillation structures of different modes; a probability density function of the traffic volume time series is fitted using Hermite polynomials with different orders according to equation (4), so that distribution characteristics of different distribution structures in the traffic volume time series are obtained through decomposition;
f(x) is set as the probability density function of the vehicle distribution, then the n-order Hermite polynomial approximation conversion is an optimization problem as follows:
{
Objective
function
:
f
^
(
x
)
=
(
∑
0
n
w
n
h
n
(
x
)
)
2
Constraint
:
∑
0
n
w
n
2
=
1
(
5
)
wherein equation (5) is solved by a constrained nonlinear optimization, and points on an N-dimensional sphere N are projected onto a plane N+1 by inverse stereographic projection, so that an unconstrained optimization solution is performed on the plane, with a specific algebraic expression of
[
P
(
ω
)
]
n
=
ω
n
(
1
-
ω
0
)
expressed as follows:
{
[
P
-
1
(
γ
)
]
n
=
{
S
2
-
1
S
2
+
1
,
if
n
=
0
2
γ
n
S
2
+
1
,
if
n
≥
1
S
2
=
∑
0
n
γ
n
2
(
6
)
maximum likelihood estimation is constructed as follows:
{
Objective
function
:
f
^
(
x
)
=
(
∑
k
=
0
n
w
k
h
k
(
x
)
)
2
Likelyhood
function
:
ℓ
(
w
)
=
∑
i
=
1
K
log
(
∑
k
=
0
n
[
P
(
w
)
]
k
h
k
(
x
i
)
)
2
.
(
7
)