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Thermodynamics


Non-equilibrium Thermodynamics & Life (2020)

Equilibrium Thermodynamics +
       Entropy Dissipation in Irreversible Process
Local Equilibrium Thermodynamics
Steady State
Dissipative Processes and Structures
Entropy and Information
Self-assembly and Self-organization
Origin of Life

Equilibrium Thermodynamics + Irreversibility

For example, the variation of the internal energy U can be expressed as the difference between the initial (i) and finial (f) states :
U = TS - PV + N ----- (2)
where U = Uf - Ui, ... Variation of heat and work are defined by Q = TS and W = PV respectively. Note that the changes are induced by the extensive variables.

A more specific example is provided by the idealized Carnot Engine Cycle (Figure 02,a): See "Statistical Physics of Self-replication" for derivation of Eq.(2a).

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Local Equilibrium Thermodynamics

Local Equilibrium Thermodynamics Diffusion In Local Equilibrium Thermodynamics, the whole system is not in equilibrium; but the intensive variables such as T, P, ... can be defined within a small volume, i.e., they are homogeneous (well-mixed) inside that volume (Figure 04,a). In addition, if the system can be described by mathematical formulas, then these intensive variables have to be differentiable, i.e., the functions should be smooth, continuous, and no break, angle, or cusp so that the derivative exists across the

Figure 04 Local Equilibrium Thermodynamics

Figure 05 Diffusion [view large image]

whole system (Figure 04,b). Accordingly, the so-called Gibbs equation in the form of Eq.2 can be carried over with the difference "" replaced by the differential "d" (see Figure 04,b) : dU = TdS - PdV + dN ----- (3).
A simple example is provided by the process of diffusion, where dU = 0, and dV = 0. In terms of entropy per unit volume s, and number density n, the rate of change of s can be derived from Eq.(3) by considering the x direction only without loss of generality :

T(x)[ds(x)/dt] = - (n,x)[dn(x)/dt] ----- (4).

By the equation of continuity, the variation in time can be related to the flux J crossing a surface (see "Fluid Dynamics") :
n/t = - (J/x) ----- (5),
where "" is the partial derivative similar to "d" but is specific for a particular variable in case there are many.

An additional assumption of local equilibrium thermodynamics is that the flux (see Onsager reciprocal relations below)
J = L(/x) ----- (6)
where L is a phenomenological coefficient (in #density-cm2/erg-sec). Finally, we obtain the diffusion equation :

----- (7),
where D = L(/n) (in dimension cm2/sec).
A more general formalism considers the intensive variable as some sort of thermodynamic force Fi, which can be or T or P, ... It has been shown that in the regime where both the flux Ji are small and the thermodynamic forces vary slowly, the rate of creation of entropy is linearly related to the fluxes :

.

The second law of thermodynamics requires that the matrix L be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that L is also symmetric, i.e., Li,j = Lj,i. It is hence known as Onsager reciprocal relations.

Applying Eq.(8) to the example of diffusion yields : s/t = L(/x)2 = [L(/n)2](n/x)2 0 ----- (9),
while it looks different from Eq.(7), the dimension is correct at least.

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Steady State

Molecules and atoms in solid state can form structure of certain shape that can last for long time. The stability is sustained by force which holds them together. In liquid or gaseous state, energy infusion from external environment melts or evaporates the solid into form-less state (suitable for fluid-dynamics and/or thermodynamic treatment). However, in some circumstances, the energy or entropy flow in such a way that some sort of structure re-appeared. The Rayleigh-Benard convection provides a good example to demonstrate how may such structure be formed.

Rayleigh-Benard Convection A Rayleigh-Bénard (R-B) system is created by heating a thin film of viscous liquid (oil, for example) from the bottom. For certain threshold of heat infusion, convective cells of regular form appear in the system (Figure 06,a). The formation arises via the upwardly buoyant force against the opposing viscous resistance as shown mathematically by the equation of motion in fluid dynamics (see Figure 06,b for configuration and pattern of convective flow).

Figure 06 Rayleigh-Benard Convection [view large image]

    There are 2 ways to calculate the entropy production in the system :

  1. The variation of entropy S = Sp - SE, where SE is the total entropy infusion from the external environment, while Sp is the total entropy generating within the system. For steady state S = 0, and thus,
    Sp = SE = dQ/T ----- (14),
    where the heat flow to the system dQ is the difference between the inflow from the heating surface and the outflow to the inner surface of the container (in contact with the liquid). The integration is over the entire surface and duration.

    The conductive heat transfer is : dQ/dAdt = -k(Tq - T)/x, where A is the cross-section area, t the time, x the thickness of the conducting plate, Tq the temperature of the heat source, and k the conductivity in erg/cm-sec-Ko.

  2. Another method is to adopt the formula in Eq.(8) for local equilibrium thermodynamics.
Entropy Spatial Distribution
    It turns out that the 2 methods to calculate entropy production are essentially in agreement for wide range of Ra (Figure 09,b) except:

  • When the experiment (numerically in Figure 09) turns on initially, there is substantial difference until the system stabilizes to steady flow. The effect is attributed to the assumption of local equilibrium in computing Sg. Such assumption is not valid in a complex, unsteady state (see Figure 09,a).

Figure 08 Entropy Spatial Distribution, Ra = 20100 [view large image]

Figure 08 shows the entropy spatial distribution from two-vortex numerical simulation with Ra = 20100. The minima occur at the locations of the vertexes, i.e., there's more orderly formation at those spots.
Rayleigh Number
  • Figure 09,b shows increasing entropy production as the value of Ra goes up. The insert displays the entropy production difference (Sp - Sg), which increases with Ra. The effect is also attributed to the inadequate assumption of local equilibrium.
  • Figure 09 Benard Cell Entropy Production [view large image]

    See "The Character of Entropy Production in Rayleigh–Bénard Convection".

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    Dissipative Processes and Structures

    Table 01 lists the dissipative structures in steady state over the eons as the energy flow per unit mass increases (see Figure 11,a).

    System Source m Energy Flow (erg/s-g) Comments
    First Galaxy GM/R 1.5 M = 1012Msun, R = 10 kpc, contraction time ~ 300 My
    Sun (ongoing) Solar Constant 2 m = (Solar constant) x (4D2/Msun), D = 1 AU
    Earth GM/R 200 Accretion time of planetesimals ~ 100 yrs, see "Origin of Solar System"
    Small Plant (ongoing) Solar Constant 103 100 cm2 leaf area, mass ~ 100 gm; useful light ~ 30%, efficiency ~ 3%
    Human (ongoing) ~ 300 watts 5x104 Mass = normal weight per person ~ 62 kg
    Human Brain (ongoing) 20 watts 2x105 Human Brain weight ~ 1 kg
    2000-watt Society 2000-watts 3x105 Mass = normal weight per person ~ 62 kg
    Cyanobacteria (ongoing) Solar Constant 109 For density ~ 1 g/cm3, m ~ Flux (A/V) x 0.01(efficiency), r ~ 10-4 cm

    Table 01 Dissipative Structures

    Constants and unit conversions : Solar Constant = 1.36 kW/m2, watt = joule/sec = 107 erg/sec, 1 day = 86400 sec, 1 year ~ 3x107 sec, Gravitational constant G = 6.67x10-8 cm3/sec2-gm, Msun = 2x1033 gm, Rsun = 1.3x107 cm, 1 AU = 1.5x1013 cm, and 1 pc = 3x1018 cm.
    BTW, the ultimate source of energy from the Sun is the thermo-nuclear fusion at its core.

    Europa That brings up some ideas about life in the universe. It seems that unicellular organisms such as the cyanobacteria are the earliest life form that can adjust to environmental stress readily because of simpler body plan but enough bio-functions to respond. They can live in all sorts of extremely harsh condition even in other worlds such as Europa (Figure 13e). But they would never be able to send out radio signals to announce their presence. It is no wonder that the current search for Extra-Terrestrial Life by big radio telescopes have turned up nothing so far. The chance of detecting human-like creatures is very slim as we are the product of environmental changes under sequence of unique circumstances. Statistically, it can never be reproduced.

    Figure 13e Europa
    [view large image]

    Table 02 below lists some major events about life on Earth to show the process and duration for its development with corresponding pictorial illustrations in Figure 13f.

    Event Time (GYA) Duration (GYA) Comments
    Big Bang 13.8 ~ 0 Beginning of the Universe
    Solar System 4.5 9.3 Formation of the Earth
    LUCA, ancestor of prokaryotes 3.6 0.9 Hydrosphere + Chemical reactions + RNA world
    Cyanobacteria 2.8 0.8 Refinement of primitive life + Creation of oxygenic atmosphere
    Eukaryotes 1.6 1.2 Novelty by mixing components or whole from prokaryotes
    Cambrian Explosion 0.54 1.06 From uni- to multi-cellular, cause not clear (see Figure 13g)
    Present 0 0.54 Darwinian evolution with adoption to changing environment

    Table 02 Major Events of Life on Earth

                                        

    Figure 13f History of Life on Earth [view large image]

                                        

    Figure 13g Cambrian Expolsion (???) [view large image]

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    Entropy and Information

    When expressing thermodynamics in terms of statistical mechanics, the temperature T, pressure P, ... are considered to be macro-states. There could be many micro-states within each macro-state as long as they don't change the property of that macro-state. For exmaple in Figure 13f, there are 8 particles distinguishable by their color. The distribution of particle in each of the 2 partitions defines the macro-state A, B, C, D. The number of arrangements can be calculated according to the formula shown in the Figure. This example illustrates different significance conveyed by entropy and information because they are related to differently.

    In term of , entropy is defined as : S = kBlog2() ----- (17).
    Entropy and Information As shown in Figure 13f, the macro-state A has the lowest entropy with S = 0 for = 1. This is a very special state that need special arrangement to reach and is considered to be very orderly. While the macro-state D is the most commonly occurrence at equilibrium, there's a lot of micro-states in such configuration.

    Information is defined by : I = log2(1/) ----- (18).

    It signifies that one particular micro-state is selected out of so many others such as = 70 in macro-state D, thus conveying a very specific (a lot of) information. In other words, information creates

    Figure 13f Entropy and Information [view large image]

    order out of chaos. This is the kind of arrangement suitable for the formation of DNA molecules. It has to be a very specific arrangement out of the 5 nitrogen-bases for coding a very specific gene. Both processes occur in the life cycle as shown in Figure 02d.

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    Self-assembly and Self-organization

    As shown in Figure 14,a, self-assembly extends the scope of chemistry to weaker interactions, thus encompassing more building blocks including those in nano- and other length scales. Self-assemble of parts into whole can be explained by minimizing the Gibbs free energy (Figure 14b). The dissipated energy is absorbed by the external environment. Water is an exceptionally good medium to remove such
    Information and Self-assembly energy and thus reducing the entropy inside. That's why many scenarios for the "origin of life" involves sea water either in tidal pool or hydrothermal mounds. Self-assembled structures can be thermodynamically very stable depending on the depth of the energy valley. The ossified bony structure in many fossils stands witness to its durability (Figure 14c).

    Figure 14 Minimum Energy, Self-assembly [view large image]

    See "OoLife, Self-assembly" and "virus" for further details.


    Self-organization also assembles parts into whole; however, it is very different from self-assembly which ends up in an equilibrium state. Off-equilibrium processing is the main feature of self-organization. A very specific example so essential to life is the process of exciting the ground state of carbon to the SP3 state upon the infusion of 2 ev energy (Figure 15,b). It promotes the carbon atoms into a tetrahedral structure, which provides stable covalent bonds with other atoms. This property of carbon accounts for the large number of known compounds. At least 80 percent of the 5 million chemical compounds registered as of the early 1980s contain carbon. The affinity of carbon for the most diverse elements does not differ very greatly - so that even the most diverse derivatives need not varying very much
    SP3 State of Carbon in energy content. This ability allows the organic world to exist in a special form of thermodynamic stability for as long as the Sun keeps supplying the 2-ev energy. This is the same process that creates all the parts for organizing into a living cell as shown in Figure 15c, and the basic reason for pumping energy into biological system to maintain metabolism and cellular structure.

    Figure 15 SP3 State of Carbon, Self-organization

    There's one problem with this kind of products. It tends to dissolve back to individual parts sooner or later, once the energy infusion is removed, i.e., back to equilibrium (see Figure 02d).
    See "OoLife, Self-reorganization" and "The Fundament of Cell Biology" for further details.

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    Origin of Life

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