Is the relative statistical weight of life containing states great enough at some point in the history of the universe to give those states a realistic probability?Whether or not abiogenesis and evolution are facts, there is one argument that the anti-evolution ID theorists are ill-advised to use: The argument that abiogenesis and evolution violate the second law of thermodynamics. In two posts on Uncommon Descent (here and here) we find Granville Sewell still publicizing this erroneous view. The line of argument used by Sewell is based on the observation that a local increase in order can only occur if negative entropy is transmitted across the boundary of a locality into a system (that’s the equivalent of saying positive entropy is exported out of the system). Sewell identifies this negative entropy (or “negentropy”) with the input of information across the boundary of the system. Therefore in his opinion it follows that some kind of information must be injected into the subsystem from without if the ordered structures of life have any chance of coming into existence. Ergo, since abiogenesis and evolution make no recourse to the input of information (or negentropy) beyond the confines of the Earth it follows that thermodynamics prohibits the formation of organic structures.
The observation that local increases in order must be accompanied by the transmission of negentropy into the locality is, of course, correct, but Sewell seems to hold the erroneous view that this negentropy must be reified as explicit information crossing the boundary into that locality. In Sewell's view this explicit information is the only way of offsetting the extremely small probabilities associated with organic structures. He rightly observers that at the bottom of the second law are probabilities, but he neglects to say that these probabilities are conditional probabilities. True, living structures are such a tiny class of what is possible that it is clear they have a negligible absolute probability of formation. But in reality the development of living things is a conditional probability, a probability conditioned on the given physical regime of the cosmos. Given this regime here is the big question: Is the conditional probability of evolution and abiogenist raised to realistic levels by physical conditions alone? If so then the “information cost” of this high conditional probability must reside in the improbability of the physical regime that supports abiogenesis and evolution.
Now consider two scenarios:
ONE) Firstly crystallization. Speaking in absolute terms probabilities favour crystallization even less than they do biological structures; with respect to the huge mathematical space of platonic configurational possibility the high organization of crystals is very a unrepresentative arrangement of particles; the low statistical weight of crystal configurations, a lot lower, in fact, than even the statistical weight of living structures (which can be realized in enumerable ways) ensures that crystals have a very low absolute probability. However, in real physical terms the relatively tiny statistical weight of the class of crystals is more than compensated for by the constraint imposed by the laws of physics, a constraint which eliminates huge numbers of cases, thus considerably raising the conditional probability of crystallization. For under the right physical conditions crystals easily form: A solution with the right concentration, temperature and pressure results in a bit by bit construction of the crystal as atoms randomly find their place in the crystal lattice and are fixed into position. This considerable increase in local order is offset by the appropriate entropy transactions with the surroundings, thus preserving the overall upward trend in entropy.
There is one important remark about crystallisation that I would want to make here. As I have just said entropy is exported out of the crystal locality thus preserving the second law. Another way of looking at this is to think of negentropy crossing over into the locality, increasing the order of this locality. But in fact if we look at the boundary of the locality closely we find nothing resembling the import across this boundary of instructions telling the atoms how to assemble; the assembly is inherent in the information contained in the laws of physics which constrain the locality sufficiently to render an absolutely improbable configuration very probable.
TWO) Now consider the case of a seed landing in a locality and a plant consequently growing. The growth of the plant constitutes a local increase in order. As in the case of crystallization this local increase in order is offset by the appropriate entropy transactions with the surroundings thus preserving an overall upward trend in entropy. However, this system is very different from crystallization on two counts. Firstly, the plant is not a manifestation of simple order like a crystal; it is of course very ordered but it is a very complex form of order at the same time. Secondly, the elementary entropy transactions with the surroundings are not the only thing that crosses the boundary of the locality: From the outset there is the input of a seed which is a very complex piece of genetic machinery and information. Thus, this system is very much in line with Sewell’s expectation that the locality needs some fundamental informational input before a highly ordered yet very complex structure can be constructed.
We have above then two scenarios where there is a local increase in order and yet no violation of the second law of thermodynamics. Both systems represent a generalized form of “crystallisation” in the sense that an ordered structure is built up atom by atom, molecule by molecule, as randomly jiggling particles find a place in the structure and then are locked into place. But, of course, there are obvious differences between these two forms of “crystallization”. For mineral crystallization the “information” present in the local laws of physics is sufficient to “instruct” the process, whereas for the “crystallization” of a living structure the complex machinery in the seed packet is needed. For mineral crystallization the very regular and simple order of the crystal is in absolute terms highly improbable, but that improbability need only be bought at the price of a relatively elementary set of physical equations; a scenario that presents no obvious intuitive problem to us. On the other hand the growth of complex flora requires the laws of physics to be supplemented by the input of the complex structure found in a seed; once again a scenario that presents no serious intuitive problems for us.
In terms of absolute probabilities both mineral crystallization and the growth of an organism (in this case a plant) represent the appearance of highly improbable structures. But in terms of conditional probabilities both processes have high probabilities; for mineral crystallization a high conditional probability is bought at the price of the laws of physics and the given boundary conditions of the physical system. In the case of plant life the boundary conditions are supplemented by the complex seed packet that boot straps the growth of an organism. Taken together the laws of physics and the boundary conditions of these two systems have the effect of putting a very tight constraint on their respective systems, so tight in fact that in spite of the general trend of an overall increase in disorder there is a high (conditional) probability of a very unrepresentative, albeit localized, configuration making an appearance. The constraints on the systems have the effect of removing myriad mathematically possible scenarios so that amongst the remaining class of possibility, the statistical weight associated with the development of localised order is relatively high, thus considerably enhancing the probability of the formation of that local order.
So given these two scenarios, is it at least conceivable that some combination of abiogenesis and evolution can result in the “crystallization” (metaphorically speaking) of living things and yet not violate the second law of thermodynamics? The trouble is that neither of the above two scenarios do exactly what is required of abiogenesis and evolution: Although mineral crystallization only buys order from the laws of physics and simple boundary conditions, the result is a very bland and dead form of order. In contrast the “crystallization” of actual living things, in all cases we observe, supplements the laws of physics with a complex boundary condition in the form of the input of an intricate genetic machine needed to seed the formation of life. If the concept of evolution and abiogenesis are to work as conventionally envisaged then they must work less like the concept of seed growth, and more like crystal growth, where the only information resources are some simple boundary conditions and the laws of physics. The question then is this: Can life originate without the boundary condition of an initial input of complex informational machinery? Can we get ordered complexity from simple laws and algorithms?
If evolution and abiogenesis are the source of living complexity and diversity then on current theories they are a form of generalised “crystallization” resourced only by the inherent information present in some simple starting conditions and the laws of physics. It is this contention that sticks in the gullet of the anti-evolutionists; they do not believe that such is possible. Those anti-evolutionists who have followed the work of William Dembski have conflated Dembski’s otherwise valid conservation of information argument with a conservation of complexity and concluded that the complexity of life can’t come from simplicity. In contrast I hardly need point out that the militant atheists love the idea of complexity coming from simplicity, perhaps for two reasons: 1. The origin of life is then apparently sourced in “mindless simplicity” 2. The notion of complexity coming from simplicity can be extended and with a bit of imagination it might be mooted that complexity can come from simply nothing! Although the idea of getting something for nothing is very suspect it is not true that only complexity comes from complexity; as I have remarked many times in this blog, simple algorithms and laws can generate complexity, but the big question is: Can they generate living complexity?
Crystallisation works because each stage in the crystal’s formation is a stable structure; if the current crystal contains N atoms and is stable, then so is the next crystal structure of N+1 atoms and so on in an inductive way. Each structure in this succession of structures is only separated by an atom or two and thus they effectively form a connected set of stable structures in morphospace; in other words crystals are reducibly complex. This connected object in morphospace is an implication of the laws of physics. These laws act as a constraint removing such an immense class of random possibilities that relative to this much reduced class the statistical weight of an outcome containing crystals is considerably raised. The second law of thermodynamics is an outcome of random thermal agitations ensuring that a system migrates towards macro states with the greatest number of microstates (that is, macro states with the greatest statistical weight); if because of the constraint of physics the macro states with the greatest statistical weight contain localized order then that localized order will actually be favoured. Thus crystal formation does not violate the second law of thermodynamics because the laws of physics eliminate so many mathematically possible micro states, that as the system moves toward the macro state with the greatest statistical weight, it moves toward a system that includes local ordering.
Conventional abiogenesis and evolution do not assume the input of explicit information; rather the information is conjectured to be implicit in the laws of physics. Therefore the theoretical precondition of these processes is similar to that required by crystal formation; namely, a connected set of stable bio-structures in morphospace separated by gaps small enough to be jumped by random thermal agitation and/or random mutations; that is, the theoretical precondition for the “crystallization” of life is that bio-structures are reducibly complex. As for crystal formation this kind of evolution/abiogenesis does not violate the second law of thermodynamics because the physical regime is conjectured to eliminate so many mathematically possible micro states that scenarios where there is a local increase in order occupy a considerably larger relative proportion of the now constricted space of possibilities, thereby much enhancing their probability. If at this point it seems intuitively unlikely that the second law of thermodynamics would allow the formation of such complex ordered structures, we now recall the growth of a plant from a seed: Clearly thermodynamics does not prohibit the growth of a complex organism. As for crystal growth so it is for organic growth: The twin physical constraints of boundary conditions and the laws of physics eliminate so many mathematically possible micro states that the local increase in order entailed by organic growth becomes a considerably larger relative proportion of the now constricted space of possibilities, thereby much enhancing its probability. However, it is clear that plant growth depends on the boundary condition of the initial input of organized complexity in the form of a seed: But then abiogenesis and evolution also depend on an initial input of complexity; namely, a complex arrangement in morphospace whereby stable structures form a connected set, a structure that would have to be entailed by the laws of physics. This arrangement in morphospace serves a similar purpose to the complex genetic mechanisms found in a seed. This invisible mathematical structure (if it exists) tends to confound the anti-evolutionist’s expectation that the information needed to assemble complex objects can only be found reified in complex material objects.
I must qualify the foregoing by admitting that it not at all clear that the conjectured structures in morphospace required by evolution and abiogenesis actually exist, or even can exist. Moreover, it is not at all clear that the laws of physics, as we currently understand them, imply that biological structures are reducibly complex. The anti-evolutionist’s contention that biological structures are in fact irreducibly complex may still be true. But the point I’m making here concerns the second law of thermodynamics and not (ir)reducible complexity. That point is this: If biological morphospace is populated by a set of reducibly complex stable structures (that is structures separated by small increments of change) then as in crystal formation abiogenesis and evolution would not violate the second law. Moreover, the “crystallization” of life would require no explicit information directing the process to pass through the boundary of an evolving locality in order to seed it. The information required for life would, on this conjecture, be implicit in the laws of physics in as much as that those laws entail the required arrangements in morphospace.
That simple laws of physics are information laden and can potentially define something as complex as the arrangement of structures in morphospace is another concept that anti-evolutionists may find difficult accept. As I have remarked before anti-evolutionist’s often wrongly identify simple laws and algorithms with “necessity” and thus they wrongly conclude that equations can’t carry information. Moreover they tend to conflate the concepts of complexity and information. Thus they conclude that simple algorithms and laws can’t generate complexity – this is, of course, wrong as we know that simple algorithms can generate the complexity of fractals and pseudo random sequences. Even if we concede that the anti-evolutionists are right about the irreducible complexity of bio-morphs, one thing remains clear: Whether the layout in bio-morphospace entails either reducible complexity or irreducible complexity it is clear that this space of structures is itself an extremely complex object: therefore either way the physics of our world entails a morphospace with a very complex layout. So even taking on board the anti-evolutionist’s concept of irreducible complexity we find that we have to admit that simple physical laws can generate very complex objects, whether those objects favour or block evolution!
Although it is clear that Granville Sewell’s opinion about the second law of thermodynamics are wrong I would want to concede that abiogeneiss and evolution are subject to a reasonable doubt on basis that as far as I am concerned the questions surrounding (ir)reducible complexity are not settled in my mind. It’s a good thing that I’m not a professional scientist because even this expressed uncertainty is likely to be regarded as the slippery slope to the "scientific heresy" of ID. But as I have remarked before, whatever the arrangement of biological structures in morphospace may be, I have a funny feeling that this arrangement is computationally irreducible; that is, the only way we have of acquiring analytical evidence about that arrangement is to actually carry out a very long computation and there are no other analytical short cuts. Hence, the burden of evidence is thrown back on the actual computation itself; in this case the actual workings of natural prehistory as manifested by the existence or nonexistence of paleontological evidence. If my mathematical intuitions are right then as I am not a paleontologist it doesn’t look as though I can make much further progress on this subject.
The observation that local increases in order must be accompanied by the transmission of negentropy into the locality is, of course, correct, but Sewell seems to hold the erroneous view that this negentropy must be reified as explicit information crossing the boundary into that locality. In Sewell's view this explicit information is the only way of offsetting the extremely small probabilities associated with organic structures. He rightly observers that at the bottom of the second law are probabilities, but he neglects to say that these probabilities are conditional probabilities. True, living structures are such a tiny class of what is possible that it is clear they have a negligible absolute probability of formation. But in reality the development of living things is a conditional probability, a probability conditioned on the given physical regime of the cosmos. Given this regime here is the big question: Is the conditional probability of evolution and abiogenist raised to realistic levels by physical conditions alone? If so then the “information cost” of this high conditional probability must reside in the improbability of the physical regime that supports abiogenesis and evolution.
Now consider two scenarios:
ONE) Firstly crystallization. Speaking in absolute terms probabilities favour crystallization even less than they do biological structures; with respect to the huge mathematical space of platonic configurational possibility the high organization of crystals is very a unrepresentative arrangement of particles; the low statistical weight of crystal configurations, a lot lower, in fact, than even the statistical weight of living structures (which can be realized in enumerable ways) ensures that crystals have a very low absolute probability. However, in real physical terms the relatively tiny statistical weight of the class of crystals is more than compensated for by the constraint imposed by the laws of physics, a constraint which eliminates huge numbers of cases, thus considerably raising the conditional probability of crystallization. For under the right physical conditions crystals easily form: A solution with the right concentration, temperature and pressure results in a bit by bit construction of the crystal as atoms randomly find their place in the crystal lattice and are fixed into position. This considerable increase in local order is offset by the appropriate entropy transactions with the surroundings, thus preserving the overall upward trend in entropy.
There is one important remark about crystallisation that I would want to make here. As I have just said entropy is exported out of the crystal locality thus preserving the second law. Another way of looking at this is to think of negentropy crossing over into the locality, increasing the order of this locality. But in fact if we look at the boundary of the locality closely we find nothing resembling the import across this boundary of instructions telling the atoms how to assemble; the assembly is inherent in the information contained in the laws of physics which constrain the locality sufficiently to render an absolutely improbable configuration very probable.
TWO) Now consider the case of a seed landing in a locality and a plant consequently growing. The growth of the plant constitutes a local increase in order. As in the case of crystallization this local increase in order is offset by the appropriate entropy transactions with the surroundings thus preserving an overall upward trend in entropy. However, this system is very different from crystallization on two counts. Firstly, the plant is not a manifestation of simple order like a crystal; it is of course very ordered but it is a very complex form of order at the same time. Secondly, the elementary entropy transactions with the surroundings are not the only thing that crosses the boundary of the locality: From the outset there is the input of a seed which is a very complex piece of genetic machinery and information. Thus, this system is very much in line with Sewell’s expectation that the locality needs some fundamental informational input before a highly ordered yet very complex structure can be constructed.
****
We have above then two scenarios where there is a local increase in order and yet no violation of the second law of thermodynamics. Both systems represent a generalized form of “crystallisation” in the sense that an ordered structure is built up atom by atom, molecule by molecule, as randomly jiggling particles find a place in the structure and then are locked into place. But, of course, there are obvious differences between these two forms of “crystallization”. For mineral crystallization the “information” present in the local laws of physics is sufficient to “instruct” the process, whereas for the “crystallization” of a living structure the complex machinery in the seed packet is needed. For mineral crystallization the very regular and simple order of the crystal is in absolute terms highly improbable, but that improbability need only be bought at the price of a relatively elementary set of physical equations; a scenario that presents no obvious intuitive problem to us. On the other hand the growth of complex flora requires the laws of physics to be supplemented by the input of the complex structure found in a seed; once again a scenario that presents no serious intuitive problems for us.
In terms of absolute probabilities both mineral crystallization and the growth of an organism (in this case a plant) represent the appearance of highly improbable structures. But in terms of conditional probabilities both processes have high probabilities; for mineral crystallization a high conditional probability is bought at the price of the laws of physics and the given boundary conditions of the physical system. In the case of plant life the boundary conditions are supplemented by the complex seed packet that boot straps the growth of an organism. Taken together the laws of physics and the boundary conditions of these two systems have the effect of putting a very tight constraint on their respective systems, so tight in fact that in spite of the general trend of an overall increase in disorder there is a high (conditional) probability of a very unrepresentative, albeit localized, configuration making an appearance. The constraints on the systems have the effect of removing myriad mathematically possible scenarios so that amongst the remaining class of possibility, the statistical weight associated with the development of localised order is relatively high, thus considerably enhancing the probability of the formation of that local order.
So given these two scenarios, is it at least conceivable that some combination of abiogenesis and evolution can result in the “crystallization” (metaphorically speaking) of living things and yet not violate the second law of thermodynamics? The trouble is that neither of the above two scenarios do exactly what is required of abiogenesis and evolution: Although mineral crystallization only buys order from the laws of physics and simple boundary conditions, the result is a very bland and dead form of order. In contrast the “crystallization” of actual living things, in all cases we observe, supplements the laws of physics with a complex boundary condition in the form of the input of an intricate genetic machine needed to seed the formation of life. If the concept of evolution and abiogenesis are to work as conventionally envisaged then they must work less like the concept of seed growth, and more like crystal growth, where the only information resources are some simple boundary conditions and the laws of physics. The question then is this: Can life originate without the boundary condition of an initial input of complex informational machinery? Can we get ordered complexity from simple laws and algorithms?
If evolution and abiogenesis are the source of living complexity and diversity then on current theories they are a form of generalised “crystallization” resourced only by the inherent information present in some simple starting conditions and the laws of physics. It is this contention that sticks in the gullet of the anti-evolutionists; they do not believe that such is possible. Those anti-evolutionists who have followed the work of William Dembski have conflated Dembski’s otherwise valid conservation of information argument with a conservation of complexity and concluded that the complexity of life can’t come from simplicity. In contrast I hardly need point out that the militant atheists love the idea of complexity coming from simplicity, perhaps for two reasons: 1. The origin of life is then apparently sourced in “mindless simplicity” 2. The notion of complexity coming from simplicity can be extended and with a bit of imagination it might be mooted that complexity can come from simply nothing! Although the idea of getting something for nothing is very suspect it is not true that only complexity comes from complexity; as I have remarked many times in this blog, simple algorithms and laws can generate complexity, but the big question is: Can they generate living complexity?
Crystallisation works because each stage in the crystal’s formation is a stable structure; if the current crystal contains N atoms and is stable, then so is the next crystal structure of N+1 atoms and so on in an inductive way. Each structure in this succession of structures is only separated by an atom or two and thus they effectively form a connected set of stable structures in morphospace; in other words crystals are reducibly complex. This connected object in morphospace is an implication of the laws of physics. These laws act as a constraint removing such an immense class of random possibilities that relative to this much reduced class the statistical weight of an outcome containing crystals is considerably raised. The second law of thermodynamics is an outcome of random thermal agitations ensuring that a system migrates towards macro states with the greatest number of microstates (that is, macro states with the greatest statistical weight); if because of the constraint of physics the macro states with the greatest statistical weight contain localized order then that localized order will actually be favoured. Thus crystal formation does not violate the second law of thermodynamics because the laws of physics eliminate so many mathematically possible micro states, that as the system moves toward the macro state with the greatest statistical weight, it moves toward a system that includes local ordering.
Conventional abiogenesis and evolution do not assume the input of explicit information; rather the information is conjectured to be implicit in the laws of physics. Therefore the theoretical precondition of these processes is similar to that required by crystal formation; namely, a connected set of stable bio-structures in morphospace separated by gaps small enough to be jumped by random thermal agitation and/or random mutations; that is, the theoretical precondition for the “crystallization” of life is that bio-structures are reducibly complex. As for crystal formation this kind of evolution/abiogenesis does not violate the second law of thermodynamics because the physical regime is conjectured to eliminate so many mathematically possible micro states that scenarios where there is a local increase in order occupy a considerably larger relative proportion of the now constricted space of possibilities, thereby much enhancing their probability. If at this point it seems intuitively unlikely that the second law of thermodynamics would allow the formation of such complex ordered structures, we now recall the growth of a plant from a seed: Clearly thermodynamics does not prohibit the growth of a complex organism. As for crystal growth so it is for organic growth: The twin physical constraints of boundary conditions and the laws of physics eliminate so many mathematically possible micro states that the local increase in order entailed by organic growth becomes a considerably larger relative proportion of the now constricted space of possibilities, thereby much enhancing its probability. However, it is clear that plant growth depends on the boundary condition of the initial input of organized complexity in the form of a seed: But then abiogenesis and evolution also depend on an initial input of complexity; namely, a complex arrangement in morphospace whereby stable structures form a connected set, a structure that would have to be entailed by the laws of physics. This arrangement in morphospace serves a similar purpose to the complex genetic mechanisms found in a seed. This invisible mathematical structure (if it exists) tends to confound the anti-evolutionist’s expectation that the information needed to assemble complex objects can only be found reified in complex material objects.
I must qualify the foregoing by admitting that it not at all clear that the conjectured structures in morphospace required by evolution and abiogenesis actually exist, or even can exist. Moreover, it is not at all clear that the laws of physics, as we currently understand them, imply that biological structures are reducibly complex. The anti-evolutionist’s contention that biological structures are in fact irreducibly complex may still be true. But the point I’m making here concerns the second law of thermodynamics and not (ir)reducible complexity. That point is this: If biological morphospace is populated by a set of reducibly complex stable structures (that is structures separated by small increments of change) then as in crystal formation abiogenesis and evolution would not violate the second law. Moreover, the “crystallization” of life would require no explicit information directing the process to pass through the boundary of an evolving locality in order to seed it. The information required for life would, on this conjecture, be implicit in the laws of physics in as much as that those laws entail the required arrangements in morphospace.
That simple laws of physics are information laden and can potentially define something as complex as the arrangement of structures in morphospace is another concept that anti-evolutionists may find difficult accept. As I have remarked before anti-evolutionist’s often wrongly identify simple laws and algorithms with “necessity” and thus they wrongly conclude that equations can’t carry information. Moreover they tend to conflate the concepts of complexity and information. Thus they conclude that simple algorithms and laws can’t generate complexity – this is, of course, wrong as we know that simple algorithms can generate the complexity of fractals and pseudo random sequences. Even if we concede that the anti-evolutionists are right about the irreducible complexity of bio-morphs, one thing remains clear: Whether the layout in bio-morphospace entails either reducible complexity or irreducible complexity it is clear that this space of structures is itself an extremely complex object: therefore either way the physics of our world entails a morphospace with a very complex layout. So even taking on board the anti-evolutionist’s concept of irreducible complexity we find that we have to admit that simple physical laws can generate very complex objects, whether those objects favour or block evolution!
Although it is clear that Granville Sewell’s opinion about the second law of thermodynamics are wrong I would want to concede that abiogeneiss and evolution are subject to a reasonable doubt on basis that as far as I am concerned the questions surrounding (ir)reducible complexity are not settled in my mind. It’s a good thing that I’m not a professional scientist because even this expressed uncertainty is likely to be regarded as the slippery slope to the "scientific heresy" of ID. But as I have remarked before, whatever the arrangement of biological structures in morphospace may be, I have a funny feeling that this arrangement is computationally irreducible; that is, the only way we have of acquiring analytical evidence about that arrangement is to actually carry out a very long computation and there are no other analytical short cuts. Hence, the burden of evidence is thrown back on the actual computation itself; in this case the actual workings of natural prehistory as manifested by the existence or nonexistence of paleontological evidence. If my mathematical intuitions are right then as I am not a paleontologist it doesn’t look as though I can make much further progress on this subject.
Mr T Reeves thank you for an insightful article. It has been proven beyond reasonable doubt that Evolution does *not* violate the 2LOT. Excerpt from the article:
ReplyDeleteWhen heat is transferred from one object to another, there is a change in entropy associated with that transfer:
dS = Q/T
where dS is the change in entropy, Q is the quantity of heat transferred, and T is the temperature of the object.
The earth is being bathed in radiation by the sun: the sun is giving off energy, some of which is received by the earth.
The rate of energy transfer is known:
P = dQ/dt = 1.2 x 10^17 W
when you sum over the whole surface of the earth; this is how much energy the earth receives from the sun per second (i.e. a lot).
Thus we can calculate the rate of entropy production of the earth/sun system:
dS/dt = P/Te - P/Ts
where Ts is the temperature of the sun, and Te is the temperature of the earth.
Since the sun is ~20x hotter than the earth, the P/Te term is ~20x larger than the P/Ts term.
Thus
dS/dt ~ P/Te
and plugging in numerical values, P ~ 1.2 x 10^17 W and Te = 300K,
dS/dt ~ 4 x 10^14 J/K/second
The gist: dS/dt is positive, and it's HUGE - the entropy of the sun-earth system is extremely rapidly increasing with time!
(Note that this does not include the eventuality that this radiation will (though on extremely long timescales) thermalize in deep space with the 3K CMB, which results in a further increase in entropy about 100 times larger!)
Now, how much of an entropy *decrease* is associated with the evolution of life? If it's greater than the above, there is a problem. But if it's less than the above, then the second law is satisfied, in that
deltaS_total = deltaS_sun + deltaS_life > 0
This can't be done as rigorously, but estimates can be made.
However, from thermodynamics we have the result
mu/T = -dS/dN
which, rearranged, gives us
deltaS = -N*mu/T
where mu is the chemical potential for a molecule and N is the number of molecules on the earth's surface.
The reduction in entropy due to the formation of life on earth (which we may consider as an ordering of all the molecules forming the earth's biomass), relative to that of a barren earth, is thus:
deltaS_life = S_earth_with_life - S_dead_earth ~ -Nb*mu/T
where Nb is the total number of molecules in the earth's biomass.
For a typical ideal gas, mu ~ 10*kb*T where kb is Boltzmann's constant.
The total biomass of earth is Nb ~ 10^41 (about 10^15 kg). Even multiplying by a factor of 100 to account for nonliving matter that might nevertheless be crucial to the evolution of life, Nb ~ 10^43.
Plugging the numbers in (including the factor of 100), we find that
|deltaS_life| ~ 10^44 * kb
This is a HUGE value, which leads to the mistaken impression that evolution violates the second law of thermodynamics.
However, let's compare that to the earlier value we found for the rate of increase in entropy due to the sun's radiation:
dS/dt_sun = 4 x 10^14 J/K/s = 3 x 10^37 kb/s
Dividing the two by each other
deltaS_life / dS/dt_sun = 3 x 10^6 seconds = minimum time allowable for evolution of life
3 x 10^6 seconds is about a month.
Conclusion: As long as the evolution of life on earth took more than about a month, it does not violate the second law of thermodynamics.
Even YECs agree that the earth is more than one month old.
Note that this does *not* imply that the evolution of life took a month; this is a highly idealized situation and only derives a lower limit on the time using basic physics.
The actual evolution of life took much longer, by a factor of billions.
But just one month of insolation adds enough entropy to the earth to offset the entropy decrease required to form all of the biomass on earth.
Even if this were an underestimate by a factor of a thousand, then only one hundred years would have been required to offset the reduction in entropy caused by the evolution of life.