Which forces reduce entropy production?
Alfred Hubler
Alfred Hubler is the director of the Center for Complex Systems Research at the University of
Illinois at Urbana-Champaign, USA ([email protected], http://server17.howwhy.com/blog/).
A real pendulum with friction will oscillate for a while after a short push, but will eventually
come to rest close to a location where is potential energy has a minimum. If the system is closed,
i.e. without a source of energy, it will eventually stop moving at a location near a minimum of
the potential, no matter what type of friction force acts on the pendulum. This ‘variation
principle’ is a simple concept to predict the long term behavior of mechanical systems, even if
the details of the friction forces are unknown.
For many years scientist tried to find a similarly simple variation principle for systems with a
source of energy such as a periodic forcing function or a battery in electrical systems [1].
Prigogine suggested that time rate of entropy production is at a minimum at stationary states [2].
Later the concept of entropy was generalized and used to describe the dynamics and stationary
states of open dissipative systems [3-7].
Entropy and the rate of entropy production are abstract concepts. What do they represent in terms
of basic concepts of physics and which forces minimize them? If S is the entropy of a system,
then the change in entropy, ΔS, is defined to be the ratio amount of heat, Q, removed from the
system, and the temperature T, i.e. ΔS = Q/T.
For instance in an electrical resistor the electrostatic potential drops from V1 = 5Volt to V2 =
3Volt and the potential energy of the electrons is converted to heat, Q = ΔV * I, where I is the
flow of charges, the current and where ΔV = (V1 – V2). If the resistor is kept at room temperature
T=298 Kelvin, then the amount of heat removed per unit of time is Q=4 Watt, and the entropy
production per unit of time Δt is ΔS/Δt = 4Watt / 298Kelvin = 0.14 Watt/Kelvin, for a I = 2Amp
current.
Figure 1 shows another example: A sponge in a transparent pipe. A small amount of water flows
through the pipe. The pipe is slightly inclined so that the water flows from left to right. The water
flow is I = 1 gram/second. The sponge dams the water, and the water level drops by h=2cm. The
drop in gravitational potential of the water is ΔV = g h, where g is gravity. Therefore the
following amount of potential energy is converted to heat every second: Q = I ΔV = 0.0002
Watt. At room temperature the entropy created every second is ΔS/Δt = 6 10-7 Watt/Kelvin.
The next question is: What forces would change entropy production?
The potential energy of the water creates a pressure which pushes the sponge (pressure = energy
/ volume). Since the potential energy is different upstream and downstream of the sponge there is
a force F due to the pressure difference which pushes the sponge downstream. The force is
proportional to the drop in potential energy, i.e. F ~ ΔV. This force F is opposed by a friction
force between the tube and the sponge.
Similarly there is a force on electrical resistors. At the resistor the electrostatic potential drops.
Therefore there is more electric charge on the wire upstream than downstream and the repulsion
of electric charges on the wire push the resistor in the direction of the electrical current.
Resistors are defined to be devices which cause a drop in potential energy and the drop in
potential energy causes a pressure drop. Therefore there is a force acting on all resistors which is
pushing the resistor downstream.
This force on the resistor often pushes the resistor out of the way, so that the limiting state is a
low resistance state. For systems with a constant flow a small resistance means that the
dissipation is small. Consequently the heat production is reduced. If the temperature is held
constant the entropy production is reduced too. For example, the sponge in Fig. 1 may eventually
be pushed out of the tube and the entropy production drops to a much smaller value.
In summary, there are forces acting on all resistors in systems with a flow, which tend to push
the resistor out of the way. This can lead to a reduced dissipation and a reduced entropy
production. The force on the sponge is proportional to the entropy production, if the temperature
and magnitude of the flow are kept constant (ΔS/Δt ~ I * F/T). However, when the sponge
changes its location downstream, due to the pressure difference, the entropy production stays the
same as long as the sponge is inside the tube. This means an observed change in position is not
necessarily correlated with a change in entropy production.
Figure 1 A sponge in a tube dams the water flow in a slightly inclined tube. A drop in the
potential energy of the water at the dam pushes the sponge downstream.
[1] Onsager, L. (1931). "Reciprocal relations in irreversible processes, I". Physical Review 37 (4):
405–426
[2] Prigogine, I. (1945). "Modération et transformations irreversibles des systemes
ouverts". Bulletin de la Classe des Sciences., Academie Royale de Belgique 31: 600–606.
[3] Dehmer M. and Mowshowitz, A. (2011), Generalized graph entropies, Complexity 17(2):
45–50.
[4] Gell-Mann, M. and Lloyd, S. (1996), Information measures, effective complexity, and total
information, Complexity 2(1): 44-52.
[5] Crofts, A. R. (2007), Life, information, entropy, and time: Vehicles for semantic inheritance,
Complexity 13(1): 14–5
[6] Morowitz, H. and Smith, E. (2007), Energy flow and the organization of life, Complexity
13(1): 51–59.
[7] Hübler, A. and Crutchfield, J.P. (2010), Order and disorder in open systems, Complexity
16(1): 6–9.