The Relationship between
random events in time and the
Langevin equation
Chris McCollin
Nottingham Trent University
References
Cox, D.R. and Isham, V. Point Processes. Chapman and Hall. 1980.
Sato, K. Lévy Processes and Infinitely Divisible Distributions. 1999. Cambridge: Cambridge University
Press.
Langevin, P. (1908). "Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]". C. R.
Acad. Sci. (Paris) 146: 530–533. ; reviewed by D. S. Lemons & A. Gythiel: Paul Langevin’s 1908 paper "On
the Theory of Brownian Motion" [...], Am. J. Phys. 65, 1079 (1997).
McCollin, C. and Göb, R. Can we predict a heart attack (or a design fault)? The Lévy Generator Process –
A Discussion. Proceedings of the ENBIS Conference, Prague, September 2015.
McCollin, C. Redefining Maintenance Events: A Study of Load-haul Dump Machines. Proceedings of the
ENBIS Conference, Linz, September 2014.
McCollin, C. Some Examples of the Parsum Equation and What it Means. Proceedings of the ENBIS
Conference, Linz, September 2014.
McCollin, C. and Coleman, S. Historical Published Maintenance Data: What Can It Tell Us About
Reliability Modelling? Quality and Reliability International. April 2014, Vol 30, Issue 3, pp781-795.
The Langevin Equation
• Newton’s equation 𝑚𝑞 = 𝑓(𝑞) force – no random element (planets) with initial conditions
𝑞 0 , 𝑞(0)
• 𝑚𝑞 = 𝑓 𝑞 − 𝛾𝑞 + 𝜎𝜉(𝑡)
where 𝛾𝑞 is the dissipative functional force with 𝛾 as the friction
coefficient, 𝜎 is the amplitude and 𝜉(𝑡) is the fluctuating noise force (random process). This is the
Langevin equation.
• The relationship between 𝜎 and 𝛾 is 𝜎 2 = 2𝐾𝐵 𝑇𝛾 where 𝐾𝐵 is Boltzmann’s constant.
• The Langevin equation for the harmonic oscillator
• Use 𝑓 𝑞 = −𝑚𝜔02 𝑞
• So 𝑚𝑞 = −𝑚𝜔02 𝑞 − 𝛾𝑞 + 𝜎𝜉(𝑡) where 𝜉(𝑡) is the stochastic element. Hence this is a stochastic
differential equation.
• We consider only the average of 𝑞 given by 𝑞 𝑡
.
• We can differentiate and double differentiate to get 𝑞 𝑡
over all particles (sub-atomic motion).
and 𝑞 𝑡
where 𝑞 𝑡
=
1
𝑁
𝑗 𝑞𝑗 (𝑡)
Solutions to the Langevin Equation
• 𝑚 𝑞 = −𝑚𝜔02 𝑞 − 𝛾 𝑞 + 0 + 𝑝0 𝛿(𝑡) since the mean of the errors 𝜉(𝑡) is zero and
the kick to start the process is given by a delta function.
∞
• The Laplace transform of 𝑞 𝑡 is 𝑞(𝑤) = 0 𝑞 𝑡 𝑒 −𝜔𝑡 𝑑𝑡 so that 𝑞(𝜔) = 𝜔𝑞(𝜔)
and 𝑞(𝜔) = 𝜔2 𝑞(𝜔) subject to the initial conditions 𝑞 0 = 0 and 𝑞 0 =
0 respectively.
= −𝑚𝜔02 𝑞 𝜔
function is unity.
• 𝑚𝜔2 𝑞 𝜔
• 𝑞(𝑤) =
• 𝜔1,2 =
𝑝0
𝛾
𝑚
𝜔2 +𝑚𝜔+𝜔02
𝛾
− 2𝑚
±
𝛾 2
2𝑚
=
− 𝛾𝜔 𝑞 𝜔
𝑝0
𝑚
(𝜔−𝜔1 )(𝜔−𝜔2 )
𝛾
− 𝜔02 = − 2𝑚 ± Ω
=
+ 𝑝0 . 1 where the Laplace transform of a delta
𝑝0
1
𝜔1 −𝜔2 𝜔−𝜔1
𝑚
−
1
𝜔−𝜔2
General forms
𝛾 2
2𝑚
• Assume
•
•
𝑞 𝑡
𝑞 𝑡
> 𝜔02
𝑝
0
= 2𝑚Ω
𝑒
=
−𝛾𝑡
for an overdamped harmonic oscillator
𝑝
0
𝑒 Ω𝑡 − 𝑒 −Ω𝑡 = mΩ
𝑒
2𝑚
𝑝0 −𝛾𝑡
2𝑚
𝑒
𝑚
−𝛾𝑡
• =
𝑡
0
𝑞 𝑣 𝑑𝑣 =
𝛾
m(
𝑝0
𝛾2
−Ω2 )
4𝑚2
𝛾
1 − 𝑒−
2𝑚
• Let Ω = 𝜔, 2𝑚 = 𝛽, then
absement)
•
𝛾𝑡
𝛾2
m(Ω2 − 2 )
4𝑚
𝛾𝑡
𝑝0
𝑠𝑖𝑛ℎΩ𝑡 , a measure of distance
𝑐𝑜𝑠ℎΩ𝑡 − 2𝑚Ω 𝑠𝑖𝑛ℎΩ𝑡 , a measure of velocity
• For the underdamped case, Ω → iΩ and 𝑞 𝑡
• Now
2𝑚
{𝑒 −
2𝑚
(𝑐𝑜𝑠ℎΩ𝑡 +
𝑡
0
𝛾𝑡
= 𝜔0 𝑒 −
𝛾
𝛾𝑡
𝑐𝑜𝑠ℎΩ𝑡 + 2mΩ 𝑒 −
𝛾
𝑐𝑜𝑠Ω𝑡 − 2𝑚Ω 𝑠𝑖𝑛Ω𝑡
2𝑚
2𝑚
𝑠𝑖𝑛ℎΩ𝑡}𝑡0
2𝑚Ω
𝑠𝑖𝑛ℎΩ𝑡)
𝛾
𝑝
𝛽
𝑞 𝑣 𝑑𝑣 = m(𝛽20−ω2) 1 − 𝑒 −𝛽𝑡 cosh 𝜔𝑡 + 𝜔 sinh 𝜔𝑡 , a measure of placement (or
−𝜔02 𝑝0
So Work done is the integral of the force: 2 2
(𝛽 −ω )
1 − 𝑒 −𝛽𝑡 cosh 𝜔𝑡 +
𝛽
𝜔
sinh 𝜔𝑡
Generator (harmonic oscillator) examples
(Parsummed residuals to a straight line fit)
Generator (harmonic oscillator) examples
(continued)
USS Halfbeak
General Motors Diesel Generator
USS Halfbeak Generator Maintenance times
Scatterplot of Number of Events vs Cumulative Maintenance Time
80
70
Number of Events
60
50
40
30
20
10
0
0
5000
10000
15000
Cumulative event time in hours
20000
25000
Definitions
An event is defined by either a point during the operation of the internal system
where a forced rule (based on pre-supposed human knowledge of the external
environment) comes into play (the forced rule has been set typically for
safety/operational reasons) or by a degradation of the internal system due to
conditions of usage within the confines of the external environment leading to a
breakdown of an internal component which has been in resonance with a
particular (un)known part of the external environment.
An insult is an event which is due to the interaction of the current state of the
system rules (internal inconsistency) and/or an external impulse (not self induced)
which leads to immediate energy change in the system.
Parsum Residual plot of USS Halfbeak data
(Only one rule – be battle prepared)
Scaled Parsums to remove amplitude effect
Removing the exponential effect
Linearising the curve with the arc sinh function
Scatterplot of arcsinh scaled parsums vs number of events
Arcsinh Scaled Parsums
2
1
0
-1
-2
0
10
20
30
40
Number of events
50
60
70
80
Rescaled arc sinh parsums
Scatterplot of Rescaled Arc sinh parsummed residuals
Rescaled Arc sinh parsums
1.0
0.5
0.0
-0.5
-1.0
0
10
20
30
40
50
Number of events
60
70
80
Regression fit
Fitted Line Plot
Rescaled arcsinh parsums = - 1.679 + 0.07269 event number
- 0.000499 event number^2 + 0.000003 event number^3
Rescaled arcsinh parsums
3
S
R-Sq
R-Sq(adj)
2
1
0
-1
-2
0
10
20
30
40
50
Number of Events
60
70
80
0.0389188
99.9%
99.9%
Linear Regression Residuals
Minitab output for a linear fit
Model Summary
• Coefficient of determination: 99.63%
Term
Constant
Coefficient Standard error
-1.4868
Number of events 0.051999
0.0161
0.000368
T-Value
P-Value
-92.29
0.000
141.15
0.000
The Regression Equation is Scaled arcsinh parsums = -1.4868 + 0.051999 number of
events
Discussion
• The final linear equation is given by 𝑦 = ae−λN sinh 𝑏 + 𝜔𝑁 =
𝑎e−λN (𝑠𝑖𝑛ℎ𝑏 𝑐𝑜𝑠ℎ𝜔𝑁 + 𝑐𝑜𝑠ℎ𝑏 𝑠𝑖𝑛ℎ𝜔𝑁)
• 𝑦 = e−λN 𝐴𝑐𝑜𝑠ℎ𝜔𝑁 + 𝐵𝑠𝑖𝑛ℎ𝜔𝑁 = Ce−λN (𝑐𝑜𝑠ℎ𝜔𝑁 + 𝐷𝑠𝑖𝑛ℎ𝜔𝑁)
where 𝑦 are the partial sum residuals, i.e. the energy at each of the 𝑁 points.
• This is similar to the work done shown for the Langevin equation of an
𝜔02 𝑝0
overdamped harmonic oscillator: Work done + 2 2
(𝛽 −ω )
𝜔02𝑝0
= (𝛽2−ω2) 𝑒 −𝛽𝑡 cosh 𝜔𝑡 + 𝜔𝛽 sinh 𝜔𝑡
where 𝑁 replaces 𝑡 in the parsummed equation.
Past (2015) and Future (2016) Work:
LHD Hydraulic Systems Analysis
References
Cox, D.R. and Isham, V. Point Processes. Chapman and Hall. 1980.
Sato, K. Lévy Processes and Infinitely Divisible Distributions. 1999. Cambridge: Cambridge University
Press.
Langevin, P. (1908). "Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]". C. R.
Acad. Sci. (Paris) 146: 530–533. ; reviewed by D. S. Lemons & A. Gythiel: Paul Langevin’s 1908 paper "On
the Theory of Brownian Motion" [...], Am. J. Phys. 65, 1079 (1997).
McCollin, C. and Göb, R. Can we predict a heart attack (or a design fault)? The Lévy Generator Process –
A Discussion. Proceedings of the ENBIS Conference, Prague, September 2015.
McCollin, C. Redefining Maintenance Events: A Study of Load-haul Dump Machines. Proceedings of the
ENBIS Conference, Linz, September 2014.
McCollin, C. Some Examples of the Parsum Equation and What it Means. Proceedings of the ENBIS
Conference, Linz, September 2014.
McCollin, C. and Coleman, S. Historical Published Maintenance Data: What Can It Tell Us About
Reliability Modelling? Quality and Reliability International. April 2014, Vol 30, Issue 3, pp781-795.