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A universal state equation of particle gases for passenger flights in United States
Journal Pre-proof A universal state equation of particle gases for passenger flights in United States Pei-Wen Yao, Yan-Jun Wang, Chen-Ping Zhu, Fan Wu, Ming-Hua Hu, Hui-Jie Yang, Vu Duong, Chin-Kun Hu, H. Eugene Stanley
PII: DOI: Reference:
S0378-4371(19)32088-6 https://doi.org/10.1016/j.physa.2019.123748 PHYSA 123748
To appear in:
Physica A
Received date : 19 June 2019 Revised date : 5 November 2019 Please cite this article as: P.-W. Yao, Y.-J. Wang, C.-P. Zhu et al., A universal state equation of particle gases for passenger flights in United States, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123748. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Highlights (for review)
Journal Pre-proof Dear Editor,
Thanks for your efforts to consider our submission. Let me mention a
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few merits of the present manuscript for your information. 1. Investigation on air transportation from view point of statistical
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physics based on big data has not appeared. The present manuscript mines departure/arrival records of 20 years of American domestic passenger flights.
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2. At the first time, we treat with passenger flights as particle gases, and set up a van der Waals-like state equation by erecting a universal gaseous constant R valid to 20 years.
It is hopeful to generalize it to
the cases in other countries.
3. The universal form of the state equation is found by rescaling
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those for different years.
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4. A new type of virtual particle gas called "delayor gas" is defined, which follows the same form of state equation of operational flights.
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This finding will promote the research on flight delays.
Best wishes,
Chen-Ping Zhu
Journal Pre-proof *Manuscript Click here to view linked References
A universal state equation of particle gases for passenger flights in United States Pei-Wen Yao†,1 Yan-Jun Wang†,2, 3 Chen-Ping Zhu*,1 Fan Wu,2, 4 Ming-Hua Hu,2 Hui-Jie Yang,5 Vu Duong,6 Chin-Kun Hu,7, 8 and H. Eugene Stanley*9 1
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China 3 State Key Laboratory of Air Traffic Management System and Technology, Nanjing 211100, China 4 Air Traffic Management Bureau of Northwest China, Xi’an, 710082, China 5 College of Management, University of Shanghai for Science and Technology, Shanghai, 200053, China 6 Air Traffic Management Research Institute, Nanyang Technological University, Singapore 637460, Singapore 7 Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 8 Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan 9 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA (Dated: November 5, 2019)
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Flight delays have negative impacts on passengers, carriers, and airports. To reduce these unpopular influence, we need to find the statistical law of the collective behavior of passenger flights. We use a mean-field approach to analyze big data listing the departure and arrival records of all American domestic passenger flights in 20 years. We treat passenger flights as particle gases and define their dimensionless velocity, quasi-thermodynamic quantities – pressure, volume, temperature, and mole number, respectively. By introducing phenomenological parameters a and b to set up van der Waalslike state equations, we erect a universal gaseous constant R for actually operated passenger flights, their counterparts on schedule, and ”delayor gases” defined as the difference between them. We find that the attractive coefficient of ”delayor gases” positively correlates with the average delay per flight on airports. Rescaling state equations for passenger flights across all 20 years, we find a universal function. This is a significant step toward understanding flight delays and dealing with temporal big data with the tools of statistical physics.
Journal Pre-proof 2 INTRODUCTION
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Flight delays bring disorder to air traffics, degrade passenger experience, and cause economic losses to air carriers [1–6]. The topic of flight delays has attracted the attention of researchers in the fields of transportation [7–11], mathematics [12, 13], computer science [14] and management [15, 16]. Before we could alleviate flight delays, we must understand the statistical law of the collective motion of passenger flights. In this quest researchers have rarely used statistical physics to analyze big data. As a successful counterpart, physicists have the mean-field theory, the van der Waals(vdW) equation [17], (P + aµ2 /V 2 )(V − µb) = µRT , to be specific, to describe states—pressure P , volume V , and temperature T — of real gases, where µ is the mole number, aµ2 /V 2 and µb are two modification terms to the state equation of ideal gases, i.e., P V = µRT , involving intermolecular attraction and repulsion with constants a and b as gas-dependent phenomenological parameters. Besides describing real gases, vdW equation has rarely been applied to topics outside traditional physics. We here use a vdW-like equation and a mean-field approximation to describe the collective behavior of passenger flights. Instead of a theoretical model, we set up the state equation by mining the real historical departure/arrival (D/A) data of all domestic passenger flights in the United States over 20 years. We analogy the factors causing flights to delay to the interactions among molecules in the present work. In air transportation, factors such as extreme weather, flawed air traffic control, and the decrease in airport capacity can cause flight delays. Flight delays can also be caused by congestion in airports and air sectors, and in turn they can cause delays in successive flights. Airplanes land on and take off from airports that exhibit an equivalent attraction on flights. The safe distance between flights are analogous to the effects of intermolecular repulsion. To examine the interactions between flights and their surroundings, we treat flights as a kind of macroscopic molecule - like particles. We download all the D/A records of US domestic passenger flights from 1995 to 2014 from the Bureau of Transportation Statistics (BTS) [18]. Each record includes the date, air carrier, tail number of aircraft, departure airport, destination airport, actual departure time and arrival time, and scheduled departure time and arrival time. We define the dimensionless velocity of each flight using real data. In the spirit of the vdW equation for real gases, we propose quasi-thermodynamic variables of the gases of ”flight particle”, and verify the existence of an invariant R—the universal gaseous constant in the target vdW-like equation—using data mining. We also determine how the state equations of statistical physics can be applied to air transportation. Finally we rescale the fitting results with vdW-like equations over 20 years to generate a universal state equation.
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MODEL
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We model flight gases by defining dimensionless velocity of a flight. We need to consider all passenger flights in each year as a specific gas species since schedules change from year to year. We mine the arrival - departure (AD) time interval–the time difference between an aircraft’s arrival at the first airport and its departure from the second airport. Thus the AD time interval includes the time spent at both airports and the en route time between them by the same aircraft. To include the effects of all the pertinent factors, we define a dimensionless velocity v for each flight to be v=
∆ts , ∆ta
(1)
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where ∆ta is the actual time interval, and ∆ts the time interval specified in the flight schedule, i.e., v is the actual velocity over the scheduled velocity. Thus when ∆ta of a flight is equal to its corresponding ∆ts , v = 1. When ∆ta is larger than ∆ts , i.e., a flight delays, its dimensionless velocity is smaller than 1. Here the dimensionless velocity never mean the flying velocity of a flight, but encompasses the influence of all possible delay factors. Using formula (1) we examine the difference between the actual time interval and the scheduled time interval, instead of absolute velocity of the aircraft. We further propose quasi-thermodynamic quantities for the (AD) flight particle gas. Setting the number of flights N0 on 1 January 2001 as the reference, i.e., the Avogadro constant, we define the mole number of the flight particle gas to be the ratio between the AD flight number N on a single day and N0 , µ=
N N0
(2)
We use the physics definition of temperature—the average kinetic energy of a set of molecules—to describe the daily
Journal Pre-proof 3 temperature T of a flight particle gas, T =
1 N 2 Σ v . N i=1 i
(3)
(4)
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2 E = ΣN i=1 vi − Σvi vj ≤1 vi vj .
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where vi is the dimensionless velocity of AD flight i in the same day. Here mass is set to m = 1 for all flights. Using dimensionless velocity, we examine the topological distance of an air route, instead of the geometric distance between the two airports it connects, i.e., we map all existing air routes connecting all possible pairs of airports as direct lines with the length unity. We consider a uniform cylinder container with the bottom area at both ends indicating the airport number. All flight particles move along a one - dimensional route in the container. Thus the daily volume V of the flight particle gas, i.e., the volume of the container, is equal to all the working airports in US. We define the daily energy of the particle gas as
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The first term on the right-hand side of this formula is the kinetic energy. The second term is the correlation energy, and vi vj is the product of the dimensionless velocities of any two flights on the same day. When all flights are on schedule, vi = 1.0, i.e., ∆ta = ∆ts , and the flight particle gas is in the ground state with energy E = N − N (N − 1)/2. In actual operation, when some flights delayed, i.e., when ∆ta > ∆ts , another particle source of flight delay is created, and the particle gas is in the excited state with the energy higher than that in the ground state. In thermal physics the pressure and volume of a system are the general force and general displacement, respectively. Analogy to the P − V relation in inner energy, we define the pressure of a flight particle gas on a day in a certain year to be Pi = −
Ei − Er , Vi − Vr
(5)
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where Ei and Vi are the energy and volume of a flight particle gas on a given day i, respectively. We count the number q of delayed flights for each day and select the day with the lowest q value to be the reference day in a certain year, and Er and Vr are the energy and volume on the schedule for the reference day. Obviously, the definition of formula (5) is invalid for the reference day or seldom if ever the day with Vi = Vr in each year. We modify the pressure and the volume in a vdW - like equation for flight particle gas. In thermodynamics, the number of all possible pairs of molecules with an attraction between N gas molecules is proportional to n2 (n = N/V ), i.e., the square of the molecular number density. In air transportation, however, not all flights directly interact with each other since all air routes are one - dimensional. As a lowest approximation, we examine the interactions along an air route between two airports and ignore the interactions between flights in different air routes. Let L be the number of AD air routes in a single day. In the spirit of mean-field approximation, we assume that there are N/L flights in an air route on average, and that there are 21 N (N/L − 1) interacting flight pairs. The modification to N2
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the pressure is a(µ2 /L − µ/N0 ), where a is a constant that absorbs 20 since N = µN0 . Because of intermolecular repulsion, the effective space for a gaseous molecule to move is smaller than V . Here the un-enterable safe range of a flight is equivalent to the effect of a repulsive force. We thus introduce the modification constant b, and use (V − bµ) to describe the volume for the particle gas to move. Here b is the reduction in the number of airports for one mole of particle gas of passenger flights considering the definition of volume V . The pressure of flight particle gas defined in formula (5) corresponds to our data mining results roughly. We calculate the daily pressure P of the particle gas and count the number of delayed flights q. In each year, in comparison of the 50 days with the lowest P versus the 50 days with the lowest q, we find a large ratio of matching days for the two cases (see Fig. 1). When the P of the flight particle gas is lower, the delayed flights on those days tend to be fewer. Here we have filtered out all cases of vi vj > 1.0 since we deal with delayed passenger flights. Figure 1 shows that there are 17 out of 20 years with the matching ratios larger than 72 per cent. And the matching ratios in most years are above 80 percent, which indicates that the quasi-thermodynamic quantity P defined by formula (5) would be meaningful in the description of passenger flight gases. However, pressure defined here is not in consistent with that in ordinary physics which describes the collective effect of momenta - change of molecules when they hit the internal wall of a container and were rebounded by it. Here, in air transportation, flights are self - propelled particles consuming fuel. Neither mechanical energy nor mechanical momentum is conserved, which is not comparable with the case of collisions between molecules and the wall. Therefore, the routine definition of the pressure is no longer ∂E valid. We have to have the aid of definition of the pressure in canonical ensemble, i.e., P = − ∂V . However, this formula can not be used in data mining since we can not let other variables unchange from the real data. Of course, it is not effective in practice. Moreover, this formula is not applicable because definitions are related with each other in theories. Actually, with formula (5) we define the quasi-thermodynamic quantity pressure P inspired by that in
Journal Pre-proof 4 50 0.9
Matching days
40
0.94
0.92
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0.86
0.9
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0.86
0.88
0.86
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0.84 0.8 0.74
0.72
30
0.64
0.5
20
10 1994
1996
1998
2000
2002
2004
2006
2008
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2010
2012
2014
p ro
year
/L- /N )](V-b )
(a)
0
6
2
2011
0
0.4
0.6
0.8
1.0
a = 59.73
R = 8.72
b = 1
c = -5.03
1.2
1.4
[P+a(
2
4
2
[P+a(
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8
0
/L- /N )](V-b )
FIG. 1. Matching cases between the 50 days with the lowest pressure based on formula (5) and the 50 days with the fewest delayed flights. The decimals indicate the matching ratios for a specific year.
2010
2
0.8
1.0
a = 57.81
R = 8.72
b = 1
c = -5.32
1.2
1.4
1.6
T
(d)
0
6
4
0
2001
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2
0.6
0.8
1.0
a = 63.14
R = 8.72
b = 1
c = -4.88
1.2
1.4
[P+a(
2
4
2
[P+a(
4
8
/L- /N )](V-b )
0
/L- /N )](V-b )
(c) 6
(b)
6
0.6
1.6
T
8
8
1.6
1999 2
0.6
1.0
R = 8.72
b = 1
c = -4.50 1.2
1.4
T
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T
0.8
a = 72.72
FIG. 2. [P + a(µ2 /L − µ/N0 )](V − µb) vs. µT in actual operation of passenger flights.(a)2011; (b)2010; (c)2001; (d)1999. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation(formula (6)). The common slope is R = 8.72.
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canonical ensemble, instead of the use of its genuine form. The validation of it has been checked partially by the fitting ratios of the matching days with the lowest pressure to the fewest delayed flights in Fig. 1, and further checked by the fitting ratios (all above ninety percent) of later going state equations to data points of 20 years.
RESULTS
We use the quasi-thermodynamic quantities P, V, T and µ, all mined from real data and generate a vdW-like equation for flight particle gases. We set b = 1.0 because the safe range keeps unchangeable under all circumstances. Here b also serves the unit for both repulsive and attractive forces. Figure 2 shows the high fitting ratio (red lines) for the scattered points indicating daily data, and we adopt the yearly constant a in the modification term to pressure P . Then by adjusting a for each year we find the common slope R = 8.724 ± 0.004 for all 20 years and their intercepts c. We use [P + a(µ2 /L − µ/N0 )](V − bµ) as the function of µT and plot a fitting panel for each year. The constant R = 8.72 thus serves as a universal gaseous constant for flight particle gases. This enables us to obtain the vdW-like
Journal Pre-proof
/L- /N )](V-b )
/L- /N )](V-b )
5
8
6
4 0
[P +a (
0
[P +a (
(b)
2
2
6
4
a
0
= 60.30
R = 8.72
0.8
1.0
c
0
1.2
1.4
1.6
0.8
1.0
(c)
(d)
0
8
6
0
a
0
= 61.41
b = 1 2 0.8
1.0
1.2
1.4
T
R = 8.72 c
0
= -5.79
1.6
1.8
c
= -5.54 1.6
0
2007
4
a
0
= 54.13
b = 1
2 1.0
1.2
1.4
R = 8.72
c
0
1.6
T
0
0
1.4
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2
2008
4
0
0
[P +a (
0
[P +a (
2
6
R = 8.72
1.2
10
8
a = 60.99 b = 1
0
T
0
10
0
0.6
/L- /N )](V-b )
T
2010 2
= -5.31
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2
0
0
2012
b = 1
/L- /N )](V-b )
8
0
0
(a)
= -6.67 1.8
0
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FIG. 3. [P0 + a0 (µ2 /L − µ/N0 )](V − µb) vs. µT0 on schedules of passenger flights.(a)2012; (b)2010; (c)2008; (d)2007. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation. The common slope is R = 8.72.
TABLE I. Attractive constants a and intercepts c of fitting lines of flight particle gases of passenger flights in actual operation from 1995 to 2014 in US. year a c year a c
2014 59.36 -4.68 2004 50.77 -6.03
2013 56.82 -5.15 2003 55.29 -5.45
2012 57.36 -5.08 2002 71.51 -4.47
2011 59.73 -5.03 2001 63.14 -4.88
2010 57.81 -5.32 2000 74.04 -4.51
2009 58.59 -5.34 1999 72.72 -4.50
2008 59.74 -5.55 1998 71.49 -4.54
2007 52.63 -6.40 1997 69.72 -4.78
2006 50.33 -6.10 1996 73.23 -4.46
2005 50.40 -6.12 1995 70.16 -4.64
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phenomenological equation that describes the collective behavior of actual passenger flights, i.e., µ µ2 )](V − µb) = µRT + c − L N0
(6)
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[P + a(
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Figure 2 shows the typical results. Values in both horizonal and vertical directions are obtained by dividing both sides of formula (6) by 1010 . Table 1 lists the values of a and c for all years from 1995 to 2014. For flights on-schedule, our data mining supports the same form of state equations with the same R. We denote their pressure P0 , and the attractive constant a0 . Here V , L, and µ are the same in both actual and scheduled operations because there is a one-to-one mapping between them in the primary data. Thus T0 = 1.0 by definition. Figure 3 shows examples of fitting results with real schedule data. We now do iteration to mine the daily attractive constants a′ . Inserting the daily quasi-thermodynamic quantities P , V , T , µ, the air route number L mined from real data, the same R and c from Fig. 2, and b = 1.0 into formula (6), we obtain the daily constants a′ by requiring all states to obey it, which leads to [P + a′ (
µ2 µ )](V − µb) = µRT + c − L N0
(7)
Exactly the same iteration process yields a′0 in the vdW - like equation for flight particle gases on schedule [P0 + a′0 (
µ2 µ )](V − µb) = µRT0 + c0 − L N0
(8)
where P0 is the pressure calculated using formula (5) from the particle gas on schedule under the same filtering threshold as P , i.e., vi vj ≤ 1.0, and the yearly c0 is the intercepts of the fitting lines. Note that both formulas (7)
Journal Pre-proof 6
)
1.5
0
(b)
0.0
0.00
0.05
0.10
0.20
-0.10
(c)
1.0
0.00
1.0
P+
2008 R = 8.72
-0.5
0.00
0.05
0.10
0.15
T
[
c' = -0.24
-0.05
0.20
0.05
0.10
0.15
0.20
T
(d)
0.5
0.0
2007
p ro
2
0.0
a'(
2
a'(
-0.05
/L-
0.5
[
P+
c' = -0.21
1.5
) /N )](V-b
)
0.15
T
1.5
2010 R = 8.72 -0.5
of
[
c' = -0.23
0
/N )](V-b
P+
-0.5
0.0
[
P+
2012 R = 8.72
-0.05
/L-
a'(
2
0.5
0
a'(
2
/L-
0.5
1.0 0
/N )](V-b
(a)
1.0
/L-
/N )](V-b
)
1.5
-0.5
R = 8.72
c' = -0.27
-1.0 -0.10
-0.05
0.00
0.05
0.10
0.15
0.20
T
2
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FIG. 4. [∆P + ∆a′ ( µL − Nµ0 )](V − µb) vs. µ∆T for delayor gases. (a)2012; (b)2010; (c)2008;(d)2007. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation (formula(9)). The common slope is R = 8.72.
and (8) share the same universal gaseous constant R, we now subtract (7) from (8) on both sides and obtain [∆P + ∆a′ (
µ µ2 )](V − µb) = µR∆T + ∆c′ − L N0
(9)
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where ∆P = P0 − P, ∆T = T0 − T , and ∆a′ = a′0 − a′ every day, and ∆c′ = c0 − c are the fitting intercepts for different years. Figure 4 shows the fitting results between the rectified data and formula (9). This indicates that the difference between actual and scheduled flights also follows the state equations for certain kinds of virtual particle gases which are nominated as ”delayor gases”. The passenger flights on schedule attached by them have to get into the excited states of the operational flights shown by formula (4). The fittings to real data in Fig.4 look perfect, which is easy to be understood. Formula (6) is valid yearly on average, and it erects the universal gaseous constant R. Data points (blue dots) in Fig.3 belong to the same kind of such cases. However, in formulas (7), to do iteration we keep the yearly constants b and c invariant, inversely derive out phenomenological parameters a every day from daily data (P, V, T, µ) based on formula (6), and so do a0 and ∆a in its counterpart formula (8) and (9). Therefore, three formulas are exactly valid daily by the adaptation of daily phenomenological parameters. And the perfect data-fitting in Fig.4 is due to the exactness of formula (7) and (8), in contrast with the cases in both Fig.2 and Fig.3. It also means that the difference between formula (7) and (8) has eliminate the influence of the schedule, and the delayor gases satisfies formula (9) exactly. To see whether ∆a′ in the state equation (formula (9))of delayor gases is meaningful to air traffics, we calculate the summation of all delay times of all AD flights for a certain day: ∆t = Σi (∆t2 − ∆t1 )i , where ∆t1 and ∆t2 are the scheduled and actual staying time intervals at both origin and destination airports, respectively. Then ∆d = ∆t/N is the daily average delay per flight at all airports. By plotting ∆d vs. ∆a′ , we see that they are positively correlated (see Fig. 5). Thus the constant ∆a′ is the coefficient depicting airport attraction to delayor gases. Furthermore, it is the delayor gases attached to flights that cause them to delay in viewpoint of physics. Now we rescale the vdW-like equation for 20 years. Using formula (6) and the results that fit the empirical data, we find 20 fitting lines which are parallel straight lines in Figure 6(a) with the same slope R but with different intercepts c 2 (see Table 1) for specific years. Let y = [P +a( µL − Nµ0 )](V −bµ). It displays 20 linear functions y = f (µT ). Calibrating these c into h = c/cr , where cr is the intercept in 2007 chosen as the reference, we rescale both the functions y and the variable µT with h, these straight lines collapse into a single one shown in Fig.6(b) with exponents α = β = −1. This is a universal function characterizing the common feature of different yearly - valid state functions. The scaling relation is y(µT ) = h−β f (µT hα ).
(10)
We attribute the appearance of it first to the existence of the universal gaseous constant R, second to the validation of
Journal Pre-proof 7
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FIG. 5. The average delay per flight on the countrywide airports ∆d vs. ∆a′ of ”delayor gases” displays positive correlations.
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the trick to deal these non- equilibrium processes with the vdW - like state equations of equilibrium states. Moreover, scaling exponents actually characterize the universal function itself. Therefore, it is not strange for us to see that α = β = −1, since we meet a linear universal function. This is the general law of collective motion of the particle gases for US domestic passenger flights. In future research we will verify its validity for global flights in all countries. Intercepts c appear at the right-hand side of formulas (6), (7), (8), and (9). They stem from the same source. The deficiency in the left-hand side makes all these values negative, indicating a systemic error in the vdW - like state equations. We attribute the appearance of formula (10) third to the common source of the intercepts c. The temperature at the right hand-side of the state equations describes the energy of the thermal motion of particle gases. Actually there is no completely random motion here. Instead, the scheduled AD passenger flights regularly fly along 1D air routes, but they are disturbed by negative impacts at both airports and en route. Air traffic rules enable passenger flights to condensate into 1D air routes and keeps them to operate in order. However, they are not included on the left-hand side of the state equations because they cannot be quantified as yet. We also treat all airports and air routes uniformly, which is not exactly valid in real-world operations.
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DISCUSSION AND CONCLUSION
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We have considered domestic passenger flights as particle gases, and mined the dimensionless velocity for each flight based on big data, i.e., 20 years of departure and arrival records in US. We proposed temperature T , volume V , mole number µ, and pressure P as quasi-thermodynamic quantities of flight particle gases. We use mean-field approach that analogizes the vdW equation for real gases to the state equation of flight gases. By introducing molecular constants a and b correspondingly, we verify the existence of the universal gaseous constant R, and set up a vdW-like equation for flight particle gases. We use data mining and phenomenological parameters to rectify non-equilibrium effects, which enables us to describe a typical non-equilibrium process with the state equations for equilibrium states. The data mining enables us to discover a positive correlation between the attractive constant ∆a′ of delayor gases and the average delay per flight in airports countrywide. By rescaling the vdW-like equations of 20 years, we obtain a universal state equation for flight particle gases. In this way we can overview the flight delays in 20 years, and understand the intercepts of all fitting lines as a common systemic factor for latter study. Our work presents a general law for the collective motion of AD passenger flights and serves an example of the application of statistical physics to complex systems, in our case the domestic air transportation in the United States. We acknowledge financial support from NSFC under Grant No.11775111 and 61773203. Y.J. Wang is supported by the foundation U1833126, and State Key Laboratory of Air Traffic Management System and Technology under Grant No.SKLATM201707. CKH is supported by Grant MOST 107-2112-M-259 -006 and MOST 108-2112-M-259 -008. HES in Boston University Center for Polymer Studies is supported by NSF Grants PHY-1505000, CMMI-1125290, and CHE-1213217, and by DTRA Grant HDTRA1-14-1-0017.
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2010 2009
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-2 0.4
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1.4
1.6
1.8
2011 2010
8
2009 2008 2007
6
2006 2005 2004
4 2 0 -2 0.6
2003 2002 2001
= -1
2000 1999
= -1
1998 1997
h = c/c
0.8
1.0
1.2
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1.6
1996
r
1.8
1995
2.0
2.2
Th
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T
2.0
2012
(b)
p ro
1999
[P+a(
2
[P+a(
2006
2013
10
of
2011
0
(a)
2014
/L- /N )](V-b )h
8
2012
2
2013
0
/L- /N )](V-b )
2014
FIG. 6. (a) Collected fitting lines for actual operation of flight particle gases. (b)Rescaled fitting lines with h = c/cr , exponents α = −1.0, β = −1.0.
J. Ferguson, A. Q. Kara, K. Hoffman, et al. Trans. Res. C: 33: 311(2013). E. B. Peterson, K. Neels, N. Barczi, et al. J. Trans. Econ. Pol., 47: 107(2013). M. Ball, C. Barnhart, M. Dresner, et al. Nat. Cent. Excel. Avi. Ope. Res. 1(2010). A. Cook, G. Tanner, S. Anderson. Brussels: EUROCONTROL Perf. Rev. Comm. 1(2004). Joint Economic Committee of US Congress. Washington, D. C: Joint Econ. Comm., 101(2008). A. Cook, G. Tanner. EUROCONTROL Perf. Rev. Unit,1(2015). G. Santos, M. Robin. Trans. Res. B, 44(3): 392(2010). C. Y. Hsiao, M. Hansen. Trans. Res. Rec: J. Trans. Res. Board, 1951: 104(2006). P. T. R. Wang, L. A. Schaefer, L. A. Wojcik. Proc. IEEE Dig. Avi. Sys. Conf., 1(5. B. 4): 1(2003). J. T. Wong, S. C. Tsai. J. Air Trans. Manag., 23(7): 5(2012). N. Pyrgiotis, K. M. Malone, A. Odoni. Trans. Res. C, 27(2): 60(2013). M. Dunbar, G. Froyland, C. L. Wu. Trans. Sci. 46: 204(2012). M. Abdel-Aty, C. Lee, Y. Bai, et al. J. Air Trans. Manag. 13(6): 355(2007). Y. J. Liu, W. D. Cao, S. Ma. 4th Int. Conf. on Nat. Comp. IEEE Computer Society, 500(2008). S. Ahmadbeygi, A. Cohn, Y. Guan, et al. J. Air Trans. Manag. 14(5): 221(2008). Y. Tu, M. Ball, W. Jank. J. Ame. Stat. Assoc. 103(481): 112(2008). C.-K. Hu, Chinese J. Phys. 52, 1 (2014). DOI: 10.6122/CJP.52.1 http://www.bts.gov
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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†These two authors contributed equally. * email: [email protected], [email protected].
*Declaration of Interest Statement
Journal Pre-proof Declaration of Interest Statement
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The authors declare no conflict of interest in either financial or non-financial aspect.
Chen-Ping Zhu
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On behalf of all the authors.
PII: DOI: Reference:
S0378-4371(19)32088-6 https://doi.org/10.1016/j.physa.2019.123748 PHYSA 123748
To appear in:
Physica A
Received date : 19 June 2019 Revised date : 5 November 2019 Please cite this article as: P.-W. Yao, Y.-J. Wang, C.-P. Zhu et al., A universal state equation of particle gases for passenger flights in United States, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123748. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Highlights (for review)
Journal Pre-proof Dear Editor,
Thanks for your efforts to consider our submission. Let me mention a
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few merits of the present manuscript for your information. 1. Investigation on air transportation from view point of statistical
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physics based on big data has not appeared. The present manuscript mines departure/arrival records of 20 years of American domestic passenger flights.
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2. At the first time, we treat with passenger flights as particle gases, and set up a van der Waals-like state equation by erecting a universal gaseous constant R valid to 20 years.
It is hopeful to generalize it to
the cases in other countries.
3. The universal form of the state equation is found by rescaling
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those for different years.
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4. A new type of virtual particle gas called "delayor gas" is defined, which follows the same form of state equation of operational flights.
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This finding will promote the research on flight delays.
Best wishes,
Chen-Ping Zhu
Journal Pre-proof *Manuscript Click here to view linked References
A universal state equation of particle gases for passenger flights in United States Pei-Wen Yao†,1 Yan-Jun Wang†,2, 3 Chen-Ping Zhu*,1 Fan Wu,2, 4 Ming-Hua Hu,2 Hui-Jie Yang,5 Vu Duong,6 Chin-Kun Hu,7, 8 and H. Eugene Stanley*9 1
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China 3 State Key Laboratory of Air Traffic Management System and Technology, Nanjing 211100, China 4 Air Traffic Management Bureau of Northwest China, Xi’an, 710082, China 5 College of Management, University of Shanghai for Science and Technology, Shanghai, 200053, China 6 Air Traffic Management Research Institute, Nanyang Technological University, Singapore 637460, Singapore 7 Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 8 Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan 9 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA (Dated: November 5, 2019)
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Flight delays have negative impacts on passengers, carriers, and airports. To reduce these unpopular influence, we need to find the statistical law of the collective behavior of passenger flights. We use a mean-field approach to analyze big data listing the departure and arrival records of all American domestic passenger flights in 20 years. We treat passenger flights as particle gases and define their dimensionless velocity, quasi-thermodynamic quantities – pressure, volume, temperature, and mole number, respectively. By introducing phenomenological parameters a and b to set up van der Waalslike state equations, we erect a universal gaseous constant R for actually operated passenger flights, their counterparts on schedule, and ”delayor gases” defined as the difference between them. We find that the attractive coefficient of ”delayor gases” positively correlates with the average delay per flight on airports. Rescaling state equations for passenger flights across all 20 years, we find a universal function. This is a significant step toward understanding flight delays and dealing with temporal big data with the tools of statistical physics.
Journal Pre-proof 2 INTRODUCTION
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Flight delays bring disorder to air traffics, degrade passenger experience, and cause economic losses to air carriers [1–6]. The topic of flight delays has attracted the attention of researchers in the fields of transportation [7–11], mathematics [12, 13], computer science [14] and management [15, 16]. Before we could alleviate flight delays, we must understand the statistical law of the collective motion of passenger flights. In this quest researchers have rarely used statistical physics to analyze big data. As a successful counterpart, physicists have the mean-field theory, the van der Waals(vdW) equation [17], (P + aµ2 /V 2 )(V − µb) = µRT , to be specific, to describe states—pressure P , volume V , and temperature T — of real gases, where µ is the mole number, aµ2 /V 2 and µb are two modification terms to the state equation of ideal gases, i.e., P V = µRT , involving intermolecular attraction and repulsion with constants a and b as gas-dependent phenomenological parameters. Besides describing real gases, vdW equation has rarely been applied to topics outside traditional physics. We here use a vdW-like equation and a mean-field approximation to describe the collective behavior of passenger flights. Instead of a theoretical model, we set up the state equation by mining the real historical departure/arrival (D/A) data of all domestic passenger flights in the United States over 20 years. We analogy the factors causing flights to delay to the interactions among molecules in the present work. In air transportation, factors such as extreme weather, flawed air traffic control, and the decrease in airport capacity can cause flight delays. Flight delays can also be caused by congestion in airports and air sectors, and in turn they can cause delays in successive flights. Airplanes land on and take off from airports that exhibit an equivalent attraction on flights. The safe distance between flights are analogous to the effects of intermolecular repulsion. To examine the interactions between flights and their surroundings, we treat flights as a kind of macroscopic molecule - like particles. We download all the D/A records of US domestic passenger flights from 1995 to 2014 from the Bureau of Transportation Statistics (BTS) [18]. Each record includes the date, air carrier, tail number of aircraft, departure airport, destination airport, actual departure time and arrival time, and scheduled departure time and arrival time. We define the dimensionless velocity of each flight using real data. In the spirit of the vdW equation for real gases, we propose quasi-thermodynamic variables of the gases of ”flight particle”, and verify the existence of an invariant R—the universal gaseous constant in the target vdW-like equation—using data mining. We also determine how the state equations of statistical physics can be applied to air transportation. Finally we rescale the fitting results with vdW-like equations over 20 years to generate a universal state equation.
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MODEL
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We model flight gases by defining dimensionless velocity of a flight. We need to consider all passenger flights in each year as a specific gas species since schedules change from year to year. We mine the arrival - departure (AD) time interval–the time difference between an aircraft’s arrival at the first airport and its departure from the second airport. Thus the AD time interval includes the time spent at both airports and the en route time between them by the same aircraft. To include the effects of all the pertinent factors, we define a dimensionless velocity v for each flight to be v=
∆ts , ∆ta
(1)
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where ∆ta is the actual time interval, and ∆ts the time interval specified in the flight schedule, i.e., v is the actual velocity over the scheduled velocity. Thus when ∆ta of a flight is equal to its corresponding ∆ts , v = 1. When ∆ta is larger than ∆ts , i.e., a flight delays, its dimensionless velocity is smaller than 1. Here the dimensionless velocity never mean the flying velocity of a flight, but encompasses the influence of all possible delay factors. Using formula (1) we examine the difference between the actual time interval and the scheduled time interval, instead of absolute velocity of the aircraft. We further propose quasi-thermodynamic quantities for the (AD) flight particle gas. Setting the number of flights N0 on 1 January 2001 as the reference, i.e., the Avogadro constant, we define the mole number of the flight particle gas to be the ratio between the AD flight number N on a single day and N0 , µ=
N N0
(2)
We use the physics definition of temperature—the average kinetic energy of a set of molecules—to describe the daily
Journal Pre-proof 3 temperature T of a flight particle gas, T =
1 N 2 Σ v . N i=1 i
(3)
(4)
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2 E = ΣN i=1 vi − Σvi vj ≤1 vi vj .
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where vi is the dimensionless velocity of AD flight i in the same day. Here mass is set to m = 1 for all flights. Using dimensionless velocity, we examine the topological distance of an air route, instead of the geometric distance between the two airports it connects, i.e., we map all existing air routes connecting all possible pairs of airports as direct lines with the length unity. We consider a uniform cylinder container with the bottom area at both ends indicating the airport number. All flight particles move along a one - dimensional route in the container. Thus the daily volume V of the flight particle gas, i.e., the volume of the container, is equal to all the working airports in US. We define the daily energy of the particle gas as
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The first term on the right-hand side of this formula is the kinetic energy. The second term is the correlation energy, and vi vj is the product of the dimensionless velocities of any two flights on the same day. When all flights are on schedule, vi = 1.0, i.e., ∆ta = ∆ts , and the flight particle gas is in the ground state with energy E = N − N (N − 1)/2. In actual operation, when some flights delayed, i.e., when ∆ta > ∆ts , another particle source of flight delay is created, and the particle gas is in the excited state with the energy higher than that in the ground state. In thermal physics the pressure and volume of a system are the general force and general displacement, respectively. Analogy to the P − V relation in inner energy, we define the pressure of a flight particle gas on a day in a certain year to be Pi = −
Ei − Er , Vi − Vr
(5)
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where Ei and Vi are the energy and volume of a flight particle gas on a given day i, respectively. We count the number q of delayed flights for each day and select the day with the lowest q value to be the reference day in a certain year, and Er and Vr are the energy and volume on the schedule for the reference day. Obviously, the definition of formula (5) is invalid for the reference day or seldom if ever the day with Vi = Vr in each year. We modify the pressure and the volume in a vdW - like equation for flight particle gas. In thermodynamics, the number of all possible pairs of molecules with an attraction between N gas molecules is proportional to n2 (n = N/V ), i.e., the square of the molecular number density. In air transportation, however, not all flights directly interact with each other since all air routes are one - dimensional. As a lowest approximation, we examine the interactions along an air route between two airports and ignore the interactions between flights in different air routes. Let L be the number of AD air routes in a single day. In the spirit of mean-field approximation, we assume that there are N/L flights in an air route on average, and that there are 21 N (N/L − 1) interacting flight pairs. The modification to N2
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the pressure is a(µ2 /L − µ/N0 ), where a is a constant that absorbs 20 since N = µN0 . Because of intermolecular repulsion, the effective space for a gaseous molecule to move is smaller than V . Here the un-enterable safe range of a flight is equivalent to the effect of a repulsive force. We thus introduce the modification constant b, and use (V − bµ) to describe the volume for the particle gas to move. Here b is the reduction in the number of airports for one mole of particle gas of passenger flights considering the definition of volume V . The pressure of flight particle gas defined in formula (5) corresponds to our data mining results roughly. We calculate the daily pressure P of the particle gas and count the number of delayed flights q. In each year, in comparison of the 50 days with the lowest P versus the 50 days with the lowest q, we find a large ratio of matching days for the two cases (see Fig. 1). When the P of the flight particle gas is lower, the delayed flights on those days tend to be fewer. Here we have filtered out all cases of vi vj > 1.0 since we deal with delayed passenger flights. Figure 1 shows that there are 17 out of 20 years with the matching ratios larger than 72 per cent. And the matching ratios in most years are above 80 percent, which indicates that the quasi-thermodynamic quantity P defined by formula (5) would be meaningful in the description of passenger flight gases. However, pressure defined here is not in consistent with that in ordinary physics which describes the collective effect of momenta - change of molecules when they hit the internal wall of a container and were rebounded by it. Here, in air transportation, flights are self - propelled particles consuming fuel. Neither mechanical energy nor mechanical momentum is conserved, which is not comparable with the case of collisions between molecules and the wall. Therefore, the routine definition of the pressure is no longer ∂E valid. We have to have the aid of definition of the pressure in canonical ensemble, i.e., P = − ∂V . However, this formula can not be used in data mining since we can not let other variables unchange from the real data. Of course, it is not effective in practice. Moreover, this formula is not applicable because definitions are related with each other in theories. Actually, with formula (5) we define the quasi-thermodynamic quantity pressure P inspired by that in
Journal Pre-proof 4 50 0.9
Matching days
40
0.94
0.92
0.9
0.86
0.9
0.88
0.86
0.88
0.86
0.82
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0.84 0.8 0.74
0.72
30
0.64
0.5
20
10 1994
1996
1998
2000
2002
2004
2006
2008
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2010
2012
2014
p ro
year
/L- /N )](V-b )
(a)
0
6
2
2011
0
0.4
0.6
0.8
1.0
a = 59.73
R = 8.72
b = 1
c = -5.03
1.2
1.4
[P+a(
2
4
2
[P+a(
Pr e-
8
0
/L- /N )](V-b )
FIG. 1. Matching cases between the 50 days with the lowest pressure based on formula (5) and the 50 days with the fewest delayed flights. The decimals indicate the matching ratios for a specific year.
2010
2
0.8
1.0
a = 57.81
R = 8.72
b = 1
c = -5.32
1.2
1.4
1.6
T
(d)
0
6
4
0
2001
al
2
0.6
0.8
1.0
a = 63.14
R = 8.72
b = 1
c = -4.88
1.2
1.4
[P+a(
2
4
2
[P+a(
4
8
/L- /N )](V-b )
0
/L- /N )](V-b )
(c) 6
(b)
6
0.6
1.6
T
8
8
1.6
1999 2
0.6
1.0
R = 8.72
b = 1
c = -4.50 1.2
1.4
T
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T
0.8
a = 72.72
FIG. 2. [P + a(µ2 /L − µ/N0 )](V − µb) vs. µT in actual operation of passenger flights.(a)2011; (b)2010; (c)2001; (d)1999. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation(formula (6)). The common slope is R = 8.72.
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canonical ensemble, instead of the use of its genuine form. The validation of it has been checked partially by the fitting ratios of the matching days with the lowest pressure to the fewest delayed flights in Fig. 1, and further checked by the fitting ratios (all above ninety percent) of later going state equations to data points of 20 years.
RESULTS
We use the quasi-thermodynamic quantities P, V, T and µ, all mined from real data and generate a vdW-like equation for flight particle gases. We set b = 1.0 because the safe range keeps unchangeable under all circumstances. Here b also serves the unit for both repulsive and attractive forces. Figure 2 shows the high fitting ratio (red lines) for the scattered points indicating daily data, and we adopt the yearly constant a in the modification term to pressure P . Then by adjusting a for each year we find the common slope R = 8.724 ± 0.004 for all 20 years and their intercepts c. We use [P + a(µ2 /L − µ/N0 )](V − bµ) as the function of µT and plot a fitting panel for each year. The constant R = 8.72 thus serves as a universal gaseous constant for flight particle gases. This enables us to obtain the vdW-like
Journal Pre-proof
/L- /N )](V-b )
/L- /N )](V-b )
5
8
6
4 0
[P +a (
0
[P +a (
(b)
2
2
6
4
a
0
= 60.30
R = 8.72
0.8
1.0
c
0
1.2
1.4
1.6
0.8
1.0
(c)
(d)
0
8
6
0
a
0
= 61.41
b = 1 2 0.8
1.0
1.2
1.4
T
R = 8.72 c
0
= -5.79
1.6
1.8
c
= -5.54 1.6
0
2007
4
a
0
= 54.13
b = 1
2 1.0
1.2
1.4
R = 8.72
c
0
1.6
T
0
0
1.4
p ro
2
2008
4
0
0
[P +a (
0
[P +a (
2
6
R = 8.72
1.2
10
8
a = 60.99 b = 1
0
T
0
10
0
0.6
/L- /N )](V-b )
T
2010 2
= -5.31
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2
0
0
2012
b = 1
/L- /N )](V-b )
8
0
0
(a)
= -6.67 1.8
0
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FIG. 3. [P0 + a0 (µ2 /L − µ/N0 )](V − µb) vs. µT0 on schedules of passenger flights.(a)2012; (b)2010; (c)2008; (d)2007. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation. The common slope is R = 8.72.
TABLE I. Attractive constants a and intercepts c of fitting lines of flight particle gases of passenger flights in actual operation from 1995 to 2014 in US. year a c year a c
2014 59.36 -4.68 2004 50.77 -6.03
2013 56.82 -5.15 2003 55.29 -5.45
2012 57.36 -5.08 2002 71.51 -4.47
2011 59.73 -5.03 2001 63.14 -4.88
2010 57.81 -5.32 2000 74.04 -4.51
2009 58.59 -5.34 1999 72.72 -4.50
2008 59.74 -5.55 1998 71.49 -4.54
2007 52.63 -6.40 1997 69.72 -4.78
2006 50.33 -6.10 1996 73.23 -4.46
2005 50.40 -6.12 1995 70.16 -4.64
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phenomenological equation that describes the collective behavior of actual passenger flights, i.e., µ µ2 )](V − µb) = µRT + c − L N0
(6)
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[P + a(
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Figure 2 shows the typical results. Values in both horizonal and vertical directions are obtained by dividing both sides of formula (6) by 1010 . Table 1 lists the values of a and c for all years from 1995 to 2014. For flights on-schedule, our data mining supports the same form of state equations with the same R. We denote their pressure P0 , and the attractive constant a0 . Here V , L, and µ are the same in both actual and scheduled operations because there is a one-to-one mapping between them in the primary data. Thus T0 = 1.0 by definition. Figure 3 shows examples of fitting results with real schedule data. We now do iteration to mine the daily attractive constants a′ . Inserting the daily quasi-thermodynamic quantities P , V , T , µ, the air route number L mined from real data, the same R and c from Fig. 2, and b = 1.0 into formula (6), we obtain the daily constants a′ by requiring all states to obey it, which leads to [P + a′ (
µ2 µ )](V − µb) = µRT + c − L N0
(7)
Exactly the same iteration process yields a′0 in the vdW - like equation for flight particle gases on schedule [P0 + a′0 (
µ2 µ )](V − µb) = µRT0 + c0 − L N0
(8)
where P0 is the pressure calculated using formula (5) from the particle gas on schedule under the same filtering threshold as P , i.e., vi vj ≤ 1.0, and the yearly c0 is the intercepts of the fitting lines. Note that both formulas (7)
Journal Pre-proof 6
)
1.5
0
(b)
0.0
0.00
0.05
0.10
0.20
-0.10
(c)
1.0
0.00
1.0
P+
2008 R = 8.72
-0.5
0.00
0.05
0.10
0.15
T
[
c' = -0.24
-0.05
0.20
0.05
0.10
0.15
0.20
T
(d)
0.5
0.0
2007
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2
0.0
a'(
2
a'(
-0.05
/L-
0.5
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P+
c' = -0.21
1.5
) /N )](V-b
)
0.15
T
1.5
2010 R = 8.72 -0.5
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[
c' = -0.23
0
/N )](V-b
P+
-0.5
0.0
[
P+
2012 R = 8.72
-0.05
/L-
a'(
2
0.5
0
a'(
2
/L-
0.5
1.0 0
/N )](V-b
(a)
1.0
/L-
/N )](V-b
)
1.5
-0.5
R = 8.72
c' = -0.27
-1.0 -0.10
-0.05
0.00
0.05
0.10
0.15
0.20
T
2
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FIG. 4. [∆P + ∆a′ ( µL − Nµ0 )](V − µb) vs. µ∆T for delayor gases. (a)2012; (b)2010; (c)2008;(d)2007. Blue dots represent daily values mined from historical records. Red straight lines are fitting results with vdW - like equation (formula(9)). The common slope is R = 8.72.
and (8) share the same universal gaseous constant R, we now subtract (7) from (8) on both sides and obtain [∆P + ∆a′ (
µ µ2 )](V − µb) = µR∆T + ∆c′ − L N0
(9)
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where ∆P = P0 − P, ∆T = T0 − T , and ∆a′ = a′0 − a′ every day, and ∆c′ = c0 − c are the fitting intercepts for different years. Figure 4 shows the fitting results between the rectified data and formula (9). This indicates that the difference between actual and scheduled flights also follows the state equations for certain kinds of virtual particle gases which are nominated as ”delayor gases”. The passenger flights on schedule attached by them have to get into the excited states of the operational flights shown by formula (4). The fittings to real data in Fig.4 look perfect, which is easy to be understood. Formula (6) is valid yearly on average, and it erects the universal gaseous constant R. Data points (blue dots) in Fig.3 belong to the same kind of such cases. However, in formulas (7), to do iteration we keep the yearly constants b and c invariant, inversely derive out phenomenological parameters a every day from daily data (P, V, T, µ) based on formula (6), and so do a0 and ∆a in its counterpart formula (8) and (9). Therefore, three formulas are exactly valid daily by the adaptation of daily phenomenological parameters. And the perfect data-fitting in Fig.4 is due to the exactness of formula (7) and (8), in contrast with the cases in both Fig.2 and Fig.3. It also means that the difference between formula (7) and (8) has eliminate the influence of the schedule, and the delayor gases satisfies formula (9) exactly. To see whether ∆a′ in the state equation (formula (9))of delayor gases is meaningful to air traffics, we calculate the summation of all delay times of all AD flights for a certain day: ∆t = Σi (∆t2 − ∆t1 )i , where ∆t1 and ∆t2 are the scheduled and actual staying time intervals at both origin and destination airports, respectively. Then ∆d = ∆t/N is the daily average delay per flight at all airports. By plotting ∆d vs. ∆a′ , we see that they are positively correlated (see Fig. 5). Thus the constant ∆a′ is the coefficient depicting airport attraction to delayor gases. Furthermore, it is the delayor gases attached to flights that cause them to delay in viewpoint of physics. Now we rescale the vdW-like equation for 20 years. Using formula (6) and the results that fit the empirical data, we find 20 fitting lines which are parallel straight lines in Figure 6(a) with the same slope R but with different intercepts c 2 (see Table 1) for specific years. Let y = [P +a( µL − Nµ0 )](V −bµ). It displays 20 linear functions y = f (µT ). Calibrating these c into h = c/cr , where cr is the intercept in 2007 chosen as the reference, we rescale both the functions y and the variable µT with h, these straight lines collapse into a single one shown in Fig.6(b) with exponents α = β = −1. This is a universal function characterizing the common feature of different yearly - valid state functions. The scaling relation is y(µT ) = h−β f (µT hα ).
(10)
We attribute the appearance of it first to the existence of the universal gaseous constant R, second to the validation of
Journal Pre-proof 7
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FIG. 5. The average delay per flight on the countrywide airports ∆d vs. ∆a′ of ”delayor gases” displays positive correlations.
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the trick to deal these non- equilibrium processes with the vdW - like state equations of equilibrium states. Moreover, scaling exponents actually characterize the universal function itself. Therefore, it is not strange for us to see that α = β = −1, since we meet a linear universal function. This is the general law of collective motion of the particle gases for US domestic passenger flights. In future research we will verify its validity for global flights in all countries. Intercepts c appear at the right-hand side of formulas (6), (7), (8), and (9). They stem from the same source. The deficiency in the left-hand side makes all these values negative, indicating a systemic error in the vdW - like state equations. We attribute the appearance of formula (10) third to the common source of the intercepts c. The temperature at the right hand-side of the state equations describes the energy of the thermal motion of particle gases. Actually there is no completely random motion here. Instead, the scheduled AD passenger flights regularly fly along 1D air routes, but they are disturbed by negative impacts at both airports and en route. Air traffic rules enable passenger flights to condensate into 1D air routes and keeps them to operate in order. However, they are not included on the left-hand side of the state equations because they cannot be quantified as yet. We also treat all airports and air routes uniformly, which is not exactly valid in real-world operations.
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DISCUSSION AND CONCLUSION
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We have considered domestic passenger flights as particle gases, and mined the dimensionless velocity for each flight based on big data, i.e., 20 years of departure and arrival records in US. We proposed temperature T , volume V , mole number µ, and pressure P as quasi-thermodynamic quantities of flight particle gases. We use mean-field approach that analogizes the vdW equation for real gases to the state equation of flight gases. By introducing molecular constants a and b correspondingly, we verify the existence of the universal gaseous constant R, and set up a vdW-like equation for flight particle gases. We use data mining and phenomenological parameters to rectify non-equilibrium effects, which enables us to describe a typical non-equilibrium process with the state equations for equilibrium states. The data mining enables us to discover a positive correlation between the attractive constant ∆a′ of delayor gases and the average delay per flight in airports countrywide. By rescaling the vdW-like equations of 20 years, we obtain a universal state equation for flight particle gases. In this way we can overview the flight delays in 20 years, and understand the intercepts of all fitting lines as a common systemic factor for latter study. Our work presents a general law for the collective motion of AD passenger flights and serves an example of the application of statistical physics to complex systems, in our case the domestic air transportation in the United States. We acknowledge financial support from NSFC under Grant No.11775111 and 61773203. Y.J. Wang is supported by the foundation U1833126, and State Key Laboratory of Air Traffic Management System and Technology under Grant No.SKLATM201707. CKH is supported by Grant MOST 107-2112-M-259 -006 and MOST 108-2112-M-259 -008. HES in Boston University Center for Polymer Studies is supported by NSF Grants PHY-1505000, CMMI-1125290, and CHE-1213217, and by DTRA Grant HDTRA1-14-1-0017.
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FIG. 6. (a) Collected fitting lines for actual operation of flight particle gases. (b)Rescaled fitting lines with h = c/cr , exponents α = −1.0, β = −1.0.
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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†These two authors contributed equally. * email: [email protected], [email protected].
*Declaration of Interest Statement
Journal Pre-proof Declaration of Interest Statement
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The authors declare no conflict of interest in either financial or non-financial aspect.
Chen-Ping Zhu
Jo
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On behalf of all the authors.