This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.
When posting on Brilliant:
Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.
Markdown
Appears as
*italics* or _italics_
italics
**bold** or __bold__
bold
- bulleted - list
bulleted
list
1. numbered 2. list
numbered
list
Note: you must add a full line of space before and after lists for them to show up correctly
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
Math
Appears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3
2×3
2^{34}
234
a_{i-1}
ai−1
\frac{2}{3}
32
\sqrt{2}
2
\sum_{i=1}^3
∑i=13
\sin \theta
sinθ
\boxed{123}
123
Comments
The idea that "inevitable entropy increases means inevitable increase in disorder" is a 19th century idea following the heels of Boltzman's statistical approach to entropy. The fallacy is in trying to conclude that "therefore" everything has to tend towards disorder. There are many ordinary instances one can see in everyday life where, given energy being pumped into a system, spontaneous order arises, even though the total overall disorder rises. For example, gently shaking a bottle of differently sized pebbles will sort them out, not mix them up more.
I applaud Jeremy's efforts to put this on a more firm mathematical foundation. Thumbs up for this one.
A guy named Harold Morowitz was after similar ideas and wrote a book Energy flow in biology that argues ordered states can be maintained as long as energy flows through them. The energy flux keeps the system away from equilibrium.
The relation in the image can be turned on its head. In the displayed form, it says that for a replicating system, the heat released to the environment, plus the change in internal ordering of the replicating entity must exceed the log ratio of the growth rate π(I→II) to the rate of decay π(II→I), which we can instead call g, and r.
For an example, state I could be one cell sitting in nutrient filled liquid, and state II could be the two daughter cells sitting in slightly less nutrient filled liquid.
The forward event that happens at the growth rate g is the process of one cell eating food and splitting into two daughter cells. The backward event that happens at the rate of decay r is one of the daughter cells undergoing all of the biochemical reactions in reverse, i.e. turning back into nutrients.
We can rewrite the equation as
ΔStoteβ⟨Q⟩+ΔSint=β⟨Q⟩+ΔSint≥logrg≥rg
so that
gmax=reβ⟨Q⟩+ΔSint
there are two conclusions connecting internal ordering and energy dissipation:
for two different replicators that shed the same amount of heat during replication, the one that becomes less ordered will be able to grow faster.
for two different replicators that achieve the same amount of internal ordering, the one that sheds more heat in the process will be able to grow faster
There is also a counterintuitive conclusion to be drawn regarding decay rates. Suppose two replicators have the same internal ordering, and shed the same amount of energy. The effective growth rate of the replicator is its forward rate gmax minus its decay rate r. I.e.
gmax(r)−r=r(eβ⟨Q⟩+ΔSint−1)
What this says is that the maximum possible effective growth rate is monotonically increased by increasing the decay rate. All else being equal, the more unstable a replicating system is, the more potential it has to be competitive in evolutionary settings, which may seem quite surprising. Two replicating strands of RNA with similar thermodynamics (similar reduction in entropy in forming from building blocks ΔSint, and similar heat released upon formation), the one that is less stable can have a reproductive advantage. It is interesting to think about the implication this has for various beginning of the world scenarios that claim RNA as the original life chemistry, or on other studies where RNA and protein stability are sometimes taken as a sign of viability.
Of course, like Michael said, the main message is: systems that can find a way to flow energy can hack the second law of thermodynamics to greatly increase themselves in number. That living systems could have arisen from lifeless abiotic matter is not so surprising in light of this new law.
A simple mixture of phospholypids & simple nucleic acids in warm water may automatically start showing the characteristics of life . people should do that experiment, that wont shut the intelligent designists up, but we can always hope.
It looks complicated, but the basic idea is simple. There is a law (the second law of thermodynamics) in physics that says, roughly, the amount of disorder in the universe has to increase over time. This means that in any system that is undergoing a rearrangement, energy tends will tend to spread out in the rearranged state, atoms will tend toward arrangements that are more randomly distributed. Basically that a system will always become less organized if it is changing.
This is in contrast to living matter like you, or a dog, or a plant, or algae. These systems are in constant motion and rearrangement, yet they are constantly taking in matter that is highly disorganized (a fruit smoothie, amino acids floating in water, carbon dioxide gas, ammonia, et cetera) and transforming them into ordered structures like ATP, bone, chemical gradients, brain, et cetera, which might seem to violate the second law of thermodynamics.
The common explanation for this is that even though the living matter is organizing itself (which decreases entropy), it is increasing the entropy of its environment so much that the net entropy of the universe is increasing.
What this new calculation shows is a quantitative relationship between the ordering of the living matter, the heat it releases during its organization, the entropy increase of the universe, and finally, the nature of how quickly it makes copies of itself. The more a blob of matter can release heat into the environment, the more its replication is favored by thermodynamics. So, if any kind of matter obtains the ability to make copies of itself (for instance, a molecule made of two RNA bases, that is able to bring together single RNA bases into further molecules of two RNA bases), the more heat it can liberate upon forming, the better it will do at making copies of itself.
Another way to think about the heat release is as energy flowing through the living matter. The living matter essentially uses the energy flowing through it to hack the second law and further order itself.
When many potential replicators exist in one environment, the one that can release the most heat relative to how much it self-organizes, will have a big advantage in taking over the system. Let me know if you have more questions.
Steven Hawking discusses this concept in The Grand Design, showing how the anthropic principle determines that entropy must increase because it is a precondition for our existence. I don't understand the formula shown here, but I guess it must be an advance by quantifying the relationship between speed of the dissipation of energy and speed of self-organisation. And is it an assumption that faster organisation provides an evolutionary advantage?
@Josh Silverman
–
Hey, wanted a little career guidance for choosing a particular course in physics (cosmology to be specific).
I'm in 10th grade now, what is the procedure that I should opt after completing 10th?
any particular university or college you can recommend, btw I'm from India. Any help would be grateful!!
@Bhavna Joshi
–
If you like physics and think you'd like to do it, I would focus on what kinds of books you read, and trying to take your physics game to the next level, and keeping yourself interested rather than aiming for one college or another. If you pin your hopes and dreams on a your love for physics, you'll probably end up at a fine place you'll be happy with. I have met many people who graduated from colleges most people would not consider first rate physics or math schools who impress me much more in their abilities and knowledge than some of the people I know coming out of well known, famous places. In any case, the kinds of things that top places will probably find impressive are the kinds of things you'll do if you just live out your love for physics. Definitely get a wide range of sources for everything you're learning, try to read the Feynman lectures and internalize all of the conceptual insights there are there, if you feel ambitious, see if a local university will allow you to take part in some research project. I don't really know anything about applying to schools from India, or which schools in India to suggest, maybe @Snehal Shekatkar knows better?
@Josh Silverman
–
I read books by Stephen hawking ( The grand design , theory of everything ) and also some other related to space...I try to understand as much as i can . But what are these Feynman lectures related to ? cosmology ?
and once again Thank you for reference !
@Bhavna Joshi
–
I would say that if you are entering into college, then you should start reading basic physics and mathematics books first and devote more time to basics than the popular books. I would suggest that apart from Feynman, you start studying Halliday-Resnik, 2 volume course of calculus by Courant-John and What is mathematics by Courant. If you want to pursue basic sciences in India itself, then according to me the best places are IISERs. I am in IISER Pune and I find the undergrads here have very nice environment to become future scientists.
@Bhavna Joshi
–
The feynman lectures were lectures on the character of physical law given by a brilliant and renowned physicist named Richard Feynman.. As someone who wants to become a physicst, I figured you'd at least know how to google by now >.>
Hey Josh, I am finally into the college and this is my first year. I had promised myself to study this concept and return here! And now when I am reading this, it all makes sense!! If you don't have any problem, then can we connect somewhere else so that I could ask you questions in future (maybe Facebook)? Thanks for explaining it so well :)
@Sachin Bhatia
–
Yeah, releasing heat to the environment is a big source of entropy. The number of states into which some amount of energy can be distributed is enormous.
I cn only tell that considering universe close system we r increasing decipiation energy in it that means entropy of universe goes on increasing but at certain point delta s will b zero means if we assum saturation point cn u guess what willb situation of universe ? Is it give back 2 it in sum les amt . Tym or same as deecipiation rate reply must
Ya its true...no life without energy. ..
The life on earth must be evaluated from small chemical reactions that occurs in pre life earth.. that chemical reactions gathered energy from sun and keep on creating new molecules which then assembled by then assembled to form a new life.. life is a set of chemical reactions..
Because the process has to follow the laws of nature. I think your question is analogous to asking "why does winding an old watch cause it to tick?"; namely because the mechanisms of the watch would have it tick as energy transferred away from the spring.
In the words of another analogy:
It's like a game of chess, there is an initial state and each player is motivated towards moving the pieces and developing a game. You can consider the players as entropy, and the rules of chess are the laws of nature which entropy naturally follows. In other words entropy drives cosmic evolution, if it wasn't true then nothing would ever change and life let alone a single star would exist.
Hey josh Silverman would you please help me in understanding this thermodynamics concept.
Right now I am in second year of mechanical engineering.
Please help me
@Josh Silverman
–
I am clear with all my basic concepts of system surrounding and universe but when it comes to all thermodynamic laws and stuff like entropy I can't think over it..
So what should I do?
@Sagar Suryavanshi
–
Have you done statistical mechanics yet? I find entropy very confusing to think about starting from thermodynamics, but easier to start from statistical mechanics.
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*or_italics_**bold**or__bold__paragraph 1
paragraph 2
[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"\(...\)or\[...\]to ensure proper formatting.2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
The idea that "inevitable entropy increases means inevitable increase in disorder" is a 19th century idea following the heels of Boltzman's statistical approach to entropy. The fallacy is in trying to conclude that "therefore" everything has to tend towards disorder. There are many ordinary instances one can see in everyday life where, given energy being pumped into a system, spontaneous order arises, even though the total overall disorder rises. For example, gently shaking a bottle of differently sized pebbles will sort them out, not mix them up more.
I applaud Jeremy's efforts to put this on a more firm mathematical foundation. Thumbs up for this one.
Log in to reply
A guy named Harold Morowitz was after similar ideas and wrote a book Energy flow in biology that argues ordered states can be maintained as long as energy flows through them. The energy flux keeps the system away from equilibrium.
The relation in the image can be turned on its head. In the displayed form, it says that for a replicating system, the heat released to the environment, plus the change in internal ordering of the replicating entity must exceed the log ratio of the growth rate π(I→II) to the rate of decay π(II→I), which we can instead call g, and r.
For an example, state I could be one cell sitting in nutrient filled liquid, and state II could be the two daughter cells sitting in slightly less nutrient filled liquid.
The forward event that happens at the growth rate g is the process of one cell eating food and splitting into two daughter cells. The backward event that happens at the rate of decay r is one of the daughter cells undergoing all of the biochemical reactions in reverse, i.e. turning back into nutrients.
We can rewrite the equation as
ΔStoteβ⟨Q⟩+ΔSint=β⟨Q⟩+ΔSint≥logrg≥rg
so that
gmax=reβ⟨Q⟩+ΔSint
there are two conclusions connecting internal ordering and energy dissipation:
There is also a counterintuitive conclusion to be drawn regarding decay rates. Suppose two replicators have the same internal ordering, and shed the same amount of energy. The effective growth rate of the replicator is its forward rate gmax minus its decay rate r. I.e.
gmax(r)−r=r(eβ⟨Q⟩+ΔSint−1)
What this says is that the maximum possible effective growth rate is monotonically increased by increasing the decay rate. All else being equal, the more unstable a replicating system is, the more potential it has to be competitive in evolutionary settings, which may seem quite surprising. Two replicating strands of RNA with similar thermodynamics (similar reduction in entropy in forming from building blocks ΔSint, and similar heat released upon formation), the one that is less stable can have a reproductive advantage. It is interesting to think about the implication this has for various beginning of the world scenarios that claim RNA as the original life chemistry, or on other studies where RNA and protein stability are sometimes taken as a sign of viability.
Of course, like Michael said, the main message is: systems that can find a way to flow energy can hack the second law of thermodynamics to greatly increase themselves in number. That living systems could have arisen from lifeless abiotic matter is not so surprising in light of this new law.
Log in to reply
Have you read about Ilya Prigogine's work as well?
i understand... NOTHING!!!!!
Log in to reply
Too complicated to understand. Just 12 years old!!
A simple mixture of phospholypids & simple nucleic acids in warm water may automatically start showing the characteristics of life . people should do that experiment, that wont shut the intelligent designists up, but we can always hope.
sorry but do i have the capacity to understand this being 14 yrs.. if of my level then please will you explain?
Log in to reply
It looks complicated, but the basic idea is simple. There is a law (the second law of thermodynamics) in physics that says, roughly, the amount of disorder in the universe has to increase over time. This means that in any system that is undergoing a rearrangement, energy tends will tend to spread out in the rearranged state, atoms will tend toward arrangements that are more randomly distributed. Basically that a system will always become less organized if it is changing.
This is in contrast to living matter like you, or a dog, or a plant, or algae. These systems are in constant motion and rearrangement, yet they are constantly taking in matter that is highly disorganized (a fruit smoothie, amino acids floating in water, carbon dioxide gas, ammonia, et cetera) and transforming them into ordered structures like ATP, bone, chemical gradients, brain, et cetera, which might seem to violate the second law of thermodynamics.
The common explanation for this is that even though the living matter is organizing itself (which decreases entropy), it is increasing the entropy of its environment so much that the net entropy of the universe is increasing.
What this new calculation shows is a quantitative relationship between the ordering of the living matter, the heat it releases during its organization, the entropy increase of the universe, and finally, the nature of how quickly it makes copies of itself. The more a blob of matter can release heat into the environment, the more its replication is favored by thermodynamics. So, if any kind of matter obtains the ability to make copies of itself (for instance, a molecule made of two RNA bases, that is able to bring together single RNA bases into further molecules of two RNA bases), the more heat it can liberate upon forming, the better it will do at making copies of itself.
Another way to think about the heat release is as energy flowing through the living matter. The living matter essentially uses the energy flowing through it to hack the second law and further order itself.
When many potential replicators exist in one environment, the one that can release the most heat relative to how much it self-organizes, will have a big advantage in taking over the system. Let me know if you have more questions.
Log in to reply
Steven Hawking discusses this concept in The Grand Design, showing how the anthropic principle determines that entropy must increase because it is a precondition for our existence. I don't understand the formula shown here, but I guess it must be an advance by quantifying the relationship between speed of the dissipation of energy and speed of self-organisation. And is it an assumption that faster organisation provides an evolutionary advantage?
Thank you so much.... I don't have any questions right now.....but if I had any in future may I ask you? (related to any other other topic)
Log in to reply
Yes, definitely. Don't hesitate.
Log in to reply
Hey, wanted a little career guidance for choosing a particular course in physics (cosmology to be specific). I'm in 10th grade now, what is the procedure that I should opt after completing 10th? any particular university or college you can recommend, btw I'm from India. Any help would be grateful!!
Log in to reply
If you like physics and think you'd like to do it, I would focus on what kinds of books you read, and trying to take your physics game to the next level, and keeping yourself interested rather than aiming for one college or another. If you pin your hopes and dreams on a your love for physics, you'll probably end up at a fine place you'll be happy with. I have met many people who graduated from colleges most people would not consider first rate physics or math schools who impress me much more in their abilities and knowledge than some of the people I know coming out of well known, famous places. In any case, the kinds of things that top places will probably find impressive are the kinds of things you'll do if you just live out your love for physics. Definitely get a wide range of sources for everything you're learning, try to read the Feynman lectures and internalize all of the conceptual insights there are there, if you feel ambitious, see if a local university will allow you to take part in some research project. I don't really know anything about applying to schools from India, or which schools in India to suggest, maybe @Snehal Shekatkar knows better?
Log in to reply
I read books by Stephen hawking ( The grand design , theory of everything ) and also some other related to space...I try to understand as much as i can . But what are these Feynman lectures related to ? cosmology ? and once again Thank you for reference !
Log in to reply
I would say that if you are entering into college, then you should start reading basic physics and mathematics books first and devote more time to basics than the popular books. I would suggest that apart from Feynman, you start studying Halliday-Resnik, 2 volume course of calculus by Courant-John and What is mathematics by Courant. If you want to pursue basic sciences in India itself, then according to me the best places are IISERs. I am in IISER Pune and I find the undergrads here have very nice environment to become future scientists.
The feynman lectures were lectures on the character of physical law given by a brilliant and renowned physicist named Richard Feynman.. As someone who wants to become a physicst, I figured you'd at least know how to google by now >.>
Try oxford, cambridge, standford, or cornell
Log in to reply
What qualifications should I have before applying for these universities?
Stanford even*
Hey Josh, I am finally into the college and this is my first year. I had promised myself to study this concept and return here! And now when I am reading this, it all makes sense!! If you don't have any problem, then can we connect somewhere else so that I could ask you questions in future (maybe Facebook)? Thanks for explaining it so well :)
But how is entropy of the external environment being increased? Through Heat Generation?
Log in to reply
Yeah, releasing heat to the environment is a big source of entropy. The number of states into which some amount of energy can be distributed is enormous.
I cn only tell that considering universe close system we r increasing decipiation energy in it that means entropy of universe goes on increasing but at certain point delta s will b zero means if we assum saturation point cn u guess what willb situation of universe ? Is it give back 2 it in sum les amt . Tym or same as deecipiation rate reply must
I think you should try and read some books on thermodynamics that might help you if you are keen on knowing.
Ya You will just little bit concentration needed n mee 2
Ya its true...no life without energy. .. The life on earth must be evaluated from small chemical reactions that occurs in pre life earth.. that chemical reactions gathered energy from sun and keep on creating new molecules which then assembled by then assembled to form a new life.. life is a set of chemical reactions..
its fine equation
yes
I pretty much always suspected that ever sense I learned what entropy was... I never went about trying to validate it with any maths though lol.
Log in to reply
wow... since*
But increase in entropy is an increase in the randomness of a system and how does it provoke life.
Log in to reply
Because the process has to follow the laws of nature. I think your question is analogous to asking "why does winding an old watch cause it to tick?"; namely because the mechanisms of the watch would have it tick as energy transferred away from the spring. In the words of another analogy: It's like a game of chess, there is an initial state and each player is motivated towards moving the pieces and developing a game. You can consider the players as entropy, and the rules of chess are the laws of nature which entropy naturally follows. In other words entropy drives cosmic evolution, if it wasn't true then nothing would ever change and life let alone a single star would exist.
so delta S always !=0 for T !=0 K
Please Elaborate........
Log in to reply
Can you be more specific?
Log in to reply
Hey josh Silverman would you please help me in understanding this thermodynamics concept. Right now I am in second year of mechanical engineering. Please help me
Log in to reply
Which concept?
Sure, what are the parts you do understand?
Log in to reply
I am clear with all my basic concepts of system surrounding and universe but when it comes to all thermodynamic laws and stuff like entropy I can't think over it.. So what should I do?
Log in to reply
Have you done statistical mechanics yet? I find entropy very confusing to think about starting from thermodynamics, but easier to start from statistical mechanics.