Stochastic action principle and maximum entropy
Microsoft Word - Leastaction_modified.doc
Stochastic action principle and maximum entropy  
Q. A. Wang, F. Tsobnang, S. Bangoup, F. Dzangue, A. Jeatsa and A. Le Méhauté 
Institut Supérieur des Matériaux et Mécaniques Avancées du Mans, 44 Av. Bartholdi, 
72000 Le Mans, France 
Abstract 
A stochastic action principle for stochastic dynamics is revisited. We present 
first numerical diffusion experiments showing that the diffusion path probability 
depend exponentially on average Lagrangian action ∫=
LdtA . This result is then 
used to derive an uncertainty measure defined in a way mimicking the heat or 
entropy in the first law of thermodynamics. It is shown that the path uncertainty 
(or path entropy) can be measured by the Shannon information and that the 
maximum entropy principle and the least action principle of classical mechanics 
can be unified into a concise form 0=Aδ , averaged over all possible paths of 
stochastic motion. It is argued that this action principle, hence the maximum 
entropy principle, is simply a consequence of the mechanical equilibrium 
condition extended to the case of stochastic dynamics.  
PACS numbers : 05.45.-a,05.70.Ln,02.50.-r,89.70.+c 
1) Introduction 
It is a long time conviction of scientists that the all systems in nature optimize certain 
mathematical measures in their motion. The search for such quantities has always been a 
major objective in the efforts to understand the laws of nature. One of these measures is the 
Lagrangian action considered as a most fundamental quantity in physics. The least action 
principle1 [1] has been used to derive almost all the physical laws for regular dynamics 
(classical mechanics, optics, electricity, relativity, electromagnetism, wave motion, etc.[2]). 
This achievement explain the efforts to extend the principle to irregular dynamics such as 
equilibrium thermodynamics[3], irreversible process [4], random dynamics[5][6], stochastic 
mechanics[7][8], quantum theory[9] and quantum gravity theory[10]. We notice that in most 
of these approaches, the randomness or the uncertainty (often measured by information or 
entropy) of the irregular dynamics is not considered in the optimization methods. For 
example, we often see expression such as RR δδ =  concerning the variation of a random 
variable R with an expectation R . This is incorrect because the variation of uncertainty 
aroused by the variation of the R may play important role in the dynamics.  
Another most fundamental measure, called entropy, is frequently used in variational 
methods of thermodynamics and statistics. The word "entropy" has a well known definition 
given by Clausius in the equilibrium thermodynamics. But it is also used as a measure of 
uncertainty in stochastic dynamics. In this sense, it is also referred to as "information" or 
"informational entropy". In contrast to the action principle, entropy and its optimization have 
always been a source of controversies. It has been used in different even opposite variational 
methods based on different physical understanding of the optimization. For instance, there is 
the principle of maximum thermodynamic entropy in statistical thermodynamics[11][12], the 
maximum information-entropy[13][14] in information theory, the principle of minimum 
entropy production [15] for certain nonequilibrium dynamics, and the principle of maximum 
entropy production for others[16][17]. Certain interpretation of entropy and of its evolution 
was even thought to be in conflict with the mechanical laws[18]. Notice that these laws can be 
derived from least action principle. In fact, the definition of entropy is itself a great matter of 
investigation for both equilibrium and nonequilibrium systems since the proposition of 
Boltzmann and Gibbs entropy. Concerning the maximum entropy calculus, few people still 
                                                 
1 We continue to use this term "least action principle" here considering its popularity in the scientific community, 
although we know nowadays that the term "optimal action" is more suitable because the action of a mechanical 
system can have a maximum, or a minimum, or a stationary for real paths[19]. 
contest the fact that the maximization of Shannon entropy yields the correct exponential 
distribution. But curously enough, few people are completely satisfied by the arguments of 
Jaynes and others[12][13][14] supporting the maximum entropy principle by considering 
entropy as anthropomorphic quantity and the principle as only an inference method. This 
question will be revisited to the end of the present paper.  
In view of the fundamental character of entropy in stochastic dynamics, it seems that the 
associated variation approaches must be considered as first principles and cannot be derived 
from other ones (such as least action principle) for regular dynamics where uncertainty does 
not exist at all. However, a question we asked is whether we can formulate a more general 
variation principle covering both the optimization of action for regular dynamics and the 
optimization of information-entropy for stochastic dynamics. We can imagine a mechanical 
system originally obeying least action principle and then subject to a random perturbation 
which makes the movement stochastic. For this kind of systems, we have proposed a 
stochastic action principle [20][21][22] which was originally a combination of maximum 
entropy principle (MEP) and least action principle on the basis of the following assumptions : 
1) A random Hamiltonian system can have different paths between two points in both 
configuration space and phase space. 
2) The paths are characterized uniquely by their action. 
3) The path information is measured by Shannon entropy. 
4) The path information is maximum for real process. 
This is in fact maximization of path entropy under the constraint associated with average 
action over paths (we assume the existence of this average measure). As expected, this 
variational principle leads to a path probability depending exponentially on the Lagrangian 
action of the paths and satisfying the Fokker-Planck equation of normal diffusion[21]. Some 
diffusion laws such as Fick's laws, Ohm's law, and Fourier's law can be derived from this 
probability distribution. We noticed that the above combination of two variation principles 
could be written in a concise form 0=Aδ [22], i.e., the variation of action averaged over all 
possible paths must vanish.  
However, many disadvantages exist in the above formalism. The first one is that not all 
the above physical assumptions are obvious and convincing. For example, concerning the 
path probability, another point of view[23] says that the probability should depend on the 
average energy on the paths instead of their action. The second disadvantage of that 
formalism is we used the Shannon entropy as a starting hypothesis, which limits the validity 
of the formalism. One may think that the principle is probably no more valid if the path 
uncertainty cannot be measured by the Shannon formula. The third disadvantage is that MEP 
is already a starting hypothesis, while it was expected that the work might help to understand 
why entropy goes to maximum.  
In this work, the reasoning is totally different even opposite. The only physical 
assumption we make is a stochastic action principle (SAP), i.e., 0=Aδ . The first and second 
assumptions mentioned above are not necessary because these properties will be extracted 
from experimental results. The third and fourth assumptions become purely the consequences 
of SAP. This work is limited to the classical mechanics of Hamiltonian systems for which the 
least action principle is well formulated. Neither relativistic nor quantum effects is 
considered.  
2) Stochastic dynamics of particle diffusion 
We consider a classical Hamiltonian systems moving, maybe randomly, in the 
configuration space between two points a and b. Its Hamiltonian is given by H=T+V and its 
Lagrangian by VTL −=  where T is the kinetic energy and V the potential one. The 
Lagrangian action on a given path is ∫=
LdtA  as defined in the Lagrangian mechanics. These 
definitions need sufficiently smooth dynamics at smallest time scales of observation. In 
addition, if there are random noises perturbing the motion, the energy variation due to the 
external perturbation or internal fluctuation is negligible at a time scale τ which is 
nevertheless small with respect to the observation period. Hence VTL −=  and VTH +=  
can exist, where T  and V  are kinetic and potential energies averaged over τ  such as 
TdtT . 
It is known that if there is no random forces and if the duration of motion tab= tb -ta from a 
to b is given, there is only one possible path between a and b. However, this uniqueness of 
transport path disappears if the motion is perturbed by random forces. An example is the case 
of particle diffusion in random media, where many paths between two given points are 
possible. This effect of noise can be easily demonstrated by a thought experiment in Figure 1. 
See the caption for detailed description. In this experiment, it is expected that more a path is 
different from the least action path (straight line in the figure) between a and b, less there are 
particles traveling on that path, i.e., smaller is the probability that the path is taken by the 
particles.  
Dust particles 
 h1 h2
Figure 1 
A thought experiment for the random diffusion of the dust particles falling in the 
air. At time ta, the particles fall out of the hole at point a. At time tb, certain 
particles arrive at point b. The existence of more than one path of the particles 
from a to b can be proved by the following operations. Let us open only one hole 
h1 on a wall between a and b, we will observe dust particles at point b at time tb. 
Then close the hole h1 and open another hole h2, we can still observe particles at 
point b at time tb, as illustrated by the two curves passing respectively through h1 
and h2. Another observation of this experiment is that more a path is different 
from the vertical straight line between a and b, less there are particles traveling on 
that path, i.e., smaller is the probability that the path is taken by the particles. This 
observation can be easily verified by the numerical experiment in the following 
section. 
Now let us suppose W discrete paths from a to b. Among a very large N particles leaving 
the point a, we observe Nk ones arriving at point b by the path k. Then the probability for the 
particles to take the path k is defined by 
Nkp kab =)( . The normalization is given by 
1)( =∑ kp
ab  or, in the case of continuous space, by the path integral 1)( =∫ prD ab , where r 
denotes the continuous coordinates of the paths.  
3) A numerical experiment of particle diffusion and path probability 
Does the probability 
Nkp kab =)(  really exist for each path? If it exists, how does it 
change from path to path? What are the quantities associated with the paths which determines 
the change in path probability? To answer these questions, we have carried out numerical 
experiments (Figure 2) showing the dust particles fall from a small hole a on the top of a two 
dimensional experimental box to the bottom of the box. A noise is introduced to perturb 
symmetrically in the direction of x the falling particles. We have used three kind of noises: 
Gaussian noise, uniform noise (with amplitudes uniformly distributed between -1 and 1) and 
truncated uniform noises (uniform noise with a cutoff of magnitude between -z and z where 
z<1, i.e., the probability is zero for the magnitude between –z and z).  
Figure 2 
2a: Model of the numerical experiment showing the dust particles fall from 
a small hole a onto the bottom of the experimental box. The distribution of 
particles on the bottom (represented by the vertical bars) is caused by the 
random noise (air for example) in the direction of x. 2b: An example of 
experimental results in which the falling particles are perturbed by a noise 
whose magnitude is uniformly distributed between -1 and 1 in x. The 
vertical bars are experimental result and the curve is a Gaussian distribution 
( ) dxxxN
xdNxdp )
1)()( 2
−== , where dN(x) is the particle number in 
the interval x—x+dx, N is the total number of falling particles and σ is the 
standard deviation (sd). The experiments show that the dp(x) is always 
Gaussian whatever the noise (uniform, Gaussian or other truncated uniform 
noises).  
Dust particles 
x0 x 
The observed distributions of particles are Gaussian for the three noises. The standard 
deviation of the distributions is uniquely determined by the nature of the noise (type, maximal 
magnitude, frequency etc.). This result was expected because of the finite variances of the 
used noises and of the central limit theorem saying that the attractor distribution is a normal 
(Gaussian) one if the noises (random variables) have finite variance.  
What can we conclude from this experiment of falling particles which seems to be trivial? 
First, let us suppose that the falling distance h is small so that the path y between a and 
any position x on the bottom can be considered as a straight line and the average velocity on y 
can be given by 
y  where τ  is the motion time from a to x (see Figure 2a). In this case, it is 
easy to show that the action Ax from a to x is proportional to (x-x0)2, i.e.,  
τττττ
2)(222222
2222 hmxxmh
mymmghymmghymVTAx −−=−=−=−=−=   
where 
mT =  and 
V =  are the average kinetic and potential energy, respectively. 
This analysis applies to any smooth motion provided h is small. Considering the observed 
Gaussian distribution of the falling particles in figure 2, we can write for small h 
)exp()( AxdN xη−∝  where η is a constant. The probability that a particle takes the small 
straight path from a to x is proportional to the exponential of action Ax.  
Now let us consider large h. In this case the paths may not be straight lines. But a curved 
path from a to x can be cut into small intervals at x1, x2, .... The above analysis is still valid for 
each small segment. The probability that a particle takes the path to x is then equal to the 
product of the probabilities on every segment of that path from a to x and should be 
proportional to the exponential of the total action from a to x, i.e.,  
( ) ( ) ( )AAAp axi i
iax ηηη −=∑−=∏ −∝ expexpexp  (1) 
where Ai is the action on the segment xi and Aax is the total action on a given path from a to x. 
The constant η is a characteristic of the noise and should be the same for every segment. The 
conclusion of this section is the path probability depends exponentially on action as long as 
the particle distribution on the bottom of the box is Gaussian.  
Concerning the exponential form of path probability, there is another proposal [23] 
( )kab Hkp γ−∝ exp)( , i.e., the path probability depends exponentially on the negative 
average energy. According to this probability, the most probable path has minimum average 
energy, so that for vanishing noise (regular dynamics), this minimum energy path would be 
the unique one which must also follow the least action principle. Here we have a paradox 
because the real path given by least action principle is in general not the path of minimum 
average energy.  
4) An action principle for stochastic dynamics 
Recently, the following stochastic action principle (SAP) was postulated[20][22]  : 
0=Aδ  (2) 
where AprDA abδδ ∫= )(  is the average of the variation Aδ  over all the paths. It can be 
written as follows  
pArDAprD
AprDA
where ∫= AprDA ab)(  is the ensemble average of action A, and abSδ  is defined by 
( ) pArDAAS abab δηδδηδ ∫=−= )( . (4) 
Eq.(4) makes it possible to derive Sab directly from probability distribution if the latter is 
known. Let us consider the dynamics in the section 3 that has the exponential path probability  
η−= exp1  (5) 
where ( )∫ −= ArDZ ηexp)(  is the partition function of the distribution. A trivial calculation 
tell us that abSδ is a variation of the path entropy Sab given by Shannon formula 
∫−= pprDS ababab ln)( . (6) 
Eq.(4) is a definition of entropy or information as a measure of uncertainty of random 
variable (action in the present case)[26]. It mimics the first law of thermodynamics 
dWdUdQ +=  where EpEU i
i∑==  is the average energy,  Ei is the energy of the state i with 
probability pi, dW is the work of the forces )(
∑−=  and qj is some extensive 
variables such as volume, surface, magnetic moment etc. The work can be written as 
∑ −=−=∑ ⎟
i dEdEpdqq
EpdW . So the first law becomes dEEddQ −= . We see that by 
Eq.(4) a “heat” Q is defined as the measure the randomness of action (or of any other random 
variables in general[26]). In Eq.(6), this heat” is related to the Shannon entropy since the 
probability is exponential. If the probability is not exponential, the functional of the entropy is 
probably different from the Shannon one, as discussed in [26]. 
With the help of Eqs.(2) and (5), it is easy to verify that  
App abab δηδ −=  (7) 
and  
App abab δηδ
22 −= . (8) 
From Eqs.(7) and (8), the maximum condition of pab , i.e., 0=pabδ  and 0
2 <pabδ , is 
transformed into 0=Aδ  and 02 >Aδ  if the constant η is positive, that is the least action path 
is the most probable path. On the contrary, if η is negative, we get 0=Aδ  and 02 <Aδ , the 
most probable path is a maximum action one.  
In our previous work, we have proved that the probability distribution of Eq.(5) satisfied 
the Fokker-Planck equation in configuration space. It is easy to see that[20], in the case of 
free particle, Eq.(5) gives us the transition probability of Brownian motion with 0
where m is the mass and D the diffusion constant of the Brownian particle[25].  
5) Return to the regular least action principle 
The stochastic action principle Eq.(2) should recover the usual least action principle 0=Aδ  
when the stochastic dynamics tends to regular dynamics with vanishing noise. To show this, 
let us put the probability Eq.(5) into Eq.(6), a straightforward calculation leads to 
AZSab η+=ln . (9) 
In regular dynamics, pab=1 for the path of optimal (maximal or minimal or stationary) 
action A0 and pab=0 for other paths having different actions, so that 0=abS  from Eq.(6). We 
have only one path, the integral in the partition function gives ( ) ( )0expexp)( AAqDZ ηη −=∫ −= . 
Eq.(9) yields 0AA= . On the other hand, we have ( ) 0=−= AASab δδηδ . Thus our principle 0=Aδ  
implies 00== AA δδ  or, more generally, 0=Aδ . This is the usual action principle. 
6) Stochastic action principle and maximum entropy  
Eq.(3) tells us that the SAP given by Eq.(2) implies 
0)( =− ASab ηδ . (10) 
meaning that the quantity ( )ASab γ−  should be optimized. If we add the normalization 
condition, the SAP becomes: 
0)]1)(([ =∫ −+− pqDAS abab αηδ  (11) 
which is just the usual Jaynes principle of maximum entropy. Hence Eq.(2) is equivalent to 
the Jaynes principle applied t path entropy.  
Is Eq.(2) simply a concise mathematical form of Jaynes principle associated to average 
action? Or is there something of fundamental which may help us to understand why entropy 
gets to maximum for stable or stationary distribution?    
From section 4, we understand that, in the case of equilibrium system, the variation dEi  is 
a work dW. However, in the case of regular mechanics, dW=0 is the condition of equilibrium 
meaning that the sum of all the forces acting on the system should be zero and the net torque 
taken with respect to any axis must also vanish. So it seems reasonable to take 0=dEi  as an 
equilibrium condition for stochastic equilibrium. In other words, when a random motion is in 
(global) equilibrium, the total work ∑ ⎟
i dqq
EpdW  by all the random forces 
on all the virtual increments dqj of a state variable (e.g., volume) must vanish. As a 
consequence of  the first law, 0=dEi  naturally leads to 0][ =− US ηδ , i.e., Jaynes maximum 
entropy principle associated with the average energy 0]1[ =+− αηδ US  where S is the 
thermodynamic entropy. This analysis seems to say that the maximum entropy (maximum 
randomness) is required by the mechanical equilibrium condition in stochastic situation.  
Remember that dEi  can also be written as a variation of free energy TSUF −= , i.e., 
dFdEi = . The stochastic equilibrium condition can be put into 0=dF . 
Coming back to our SAP in Eq.(2), the system is in nonequilibrium motion. If there is no 
noise, the true path satisfies 0=Aδ  and 0=
. When there is noise perturbation, 
we have[22]  
0)( =∑ ⎥
∂∫= ∫
j ab drdtr
prDdA  
(12) 
where 0≠=
t jjj
 is the random force on drj. Let ∫=
j dtft
f 1  be the time average of 
the random force fj, we obtain 
[ ] 0)( ==∑= ∫ dWtdrfprDtdA ab
jjj abab  
(13) 
where [ ] ∑=∑= ∫∫
jj abj
jjj ab dWprDdrfprDdW )()(  is the ensemble average (over all paths) of the 
time average ∫=∫=
j dtWdt
dtrdft
dW 11   and rdfdW jjj=  is the work of random force 
over the variation (deformation) rd j  of a given path. Eq.(13) means 
0=dW  (14) 
since tab is arbitrary. Eq.(14) implies that the average work of the random forces at any 
moment over any time interval and over arbitrary path deformation must vanish. This 
condition can be satisfied only when the motion is totally random, a state at which the system 
does not have privileged degrees of freedom without constraints. Indeed, it is easy to show 
that the maximum entropy with only the normalization as constraint yields totally 
equiprobable paths. This argument also holds for equilibrium systems. The vanishing work 
0==dWdEi  needs that, if there is no other constraint than the normalization, no degree of 
freedom is privileged, i.e., all microstates of the equilibrium state should be equiprobable. 
This is the state which has the maximum randomness and uncertainty.  
To summarize this section, the optimization of both equilibrium entropy and 
nonequilibrium path entropy is simply the requirement of the mechanical equilibrium 
conditions in the case of stochastic motion. There is no mystery in that. Entropy or dynamical 
randomness (uncertainty) must take the largest value for the system to reach a state where the 
total virtual work of the random forces should vanish. Entropy is not necessarily 
anthropomorphic quantity as claimed by Jaynes[14] to be able to take maximum for correct 
inference. Entropy is nothing but a measure of physical uncertainty of stochastic situation.  
Hence maximum entropy is not merely an inference principle. It is a law of physics. This is a 
major result of the present work. 
7) Concluding remarks 
We have presented numerical experiments showing the path probability distribution of 
some stochastic dynamics depends exponentially on Lagrangian action. On this basis, a 
stochastic action principle (SAP) formulated for Hamiltonian system perturbed by random 
forces is revisited. By using a new definition of statistical uncertainty measure which mimics 
the heat in the first law of equilibrium thermodynamics, it is shown that, if the path 
probability is exponential of action, the measure of path uncertainty we defined is just 
Shannon information entropy. It is also shown that the SAP yields both the Jaynes principle of 
maximum entropy and the conventional least action principle for vanishing noise. It is argued 
that the maximum entropy is the requirement of the conventional mechanical equilibrium 
condition for the motion of random systems to be stabilized, which means the total virtual 
work of random forces should vanish at any moment within any arbitrary time interval. This 
implies, in equilibrium case, 0=dEi , and in nonequilibrium case, 0== dWdA . In both cases, 
the randomness of the motion must be at maximum in order that all degrees of freedom are 
equally probable if there is no constraint. By these arguments, we try to give the maximum 
entropy principle, considered by many as only an inference principle, the status of a 
fundamental physical law. 
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