The "Burdens of Proof" of Ecological Economics

Once again S blows my mind by lending me these amazing textbooks on the condition that I read them. So, I read them. Anyway, today I got a book on Ecological Economics. That's like manna from academic heaven to me. Well, there is this essay in there by a professor out at UCSB (citation below), and I think it's very interesting; it essentially sets out a list of ecological assumptions that need to be proved. What's so cool about it is that he really "hit the nail on the head" with these assumptions: he mentions so many economical principles that I take for granted. Without further ado, here they are:

The "golden rule" (akin to the rule of entropy according to this essay): "There are no free lunches"

Subsequent rules:

1. The economical world available to the current human population is limited to earth.

2. The first law: We can never do merely one thing

3. The second law: There's no one to throw away to

4. The third law: Population x per capita impact = total human impact on environment

5. Scale effects, though sometimes compensable, are inescapeable

6. Cultural carrying capacity is inversely related to standards of living

7. The maximum is not the optimum

8. "The greatest good for the greatest numbers is nonsense"--> I love how he describes this. So I will quote it: "The theory of partial differential equations tells us that we cannot maximize for more than one variable at a time. Since the time honored utilitarian ideal is mathematical nonsense, it must be practical nonsense, also... we must decide whether we want to maximize the total number of human beings on earth, or to maximize their average-- not their total-- well being."

9. Attempts to create perfectly reliable machine human couplings are inescapeably self-defeating.

10. Thou shall not transgress thy carrying capacity.

11. Every shortage of supply is equally a longage of demand.

I really think that's a great, concise way to sum up a lot of economic assumptions: Credit to the master below:

Hardin, Garrett. 1991. Paramount Divisions in Ecological Economics. Ecological Economics. Colombia University Press. pp. 48-57.

Dead sexy

Photo of the Martian dunes.

Cold Weather Running- 11 years of tips and tricks

While I can't solve the problem below (next on my list after the Riemann Hypothesis, of course), I thought this morning of something that I wanted to post on here. You see, this morning, it was a balmy 16 degrees in Clemson, and I decided to go running. Now, I've read a lot of articles in fine running literature such as "Runner's World" that detail the proper techniques for winter running, and I've got to say, I think that experience with winter running really dictates the proper way to handle it. With this in mind, I am going to share what my 11 years of running experience have taught me about running around when it's very cold outside.

1. If you make it out the door, consider yourself victorious. Yes, you may be able to log a 70 mile week in the summer, but that's much less feasible in the winter. And if you think before heading out the door "must run 11 miles today," it's easy to say "screw it, I'll just sleep." On the other hand, aiming for a four-mile run, that's only looking at 30 minutes of activity-- you can do that easily. The first step to cold weather running is DOING IT.

2. Always overdress. I struggled with this for years. I used to wear shorts in the winter, thinking pants hampered my ability to run fast. Then I realized: cold hampers my ability to run fast. If your quads are too tight to spread out because they are cold, your stride is going to suck. As a general rule, if the weather is less than 50 degrees, I wear pants (if I am racing, I'll extend that down to about 35 degrees, but do a good warm up inside and only go outside at the last minute). The same goes for shirts. I think the best attire is "something dry fit" + "a fleece" + "a windbreaker"... you won't sweat into your fleece, you'll stay warm, and the wind will be blocked. No, a really nice jacket isn't the same as layering-- the layers make friction and air pockets that keep you warmer. Trust me.

3. A good pair of socks is hard to find. You want a pair of socks that won't get wet, but are really thick. So thick they feel funny on your feet. I like skiing socks. If your feet get cold, you'll get cold. Believe me.

4. Mittens>Nothing>Gloves. Gloves suck. Don't use them. When you have nothing on your hands, you'll pull your hands into your sleeves and your fingers will touch, keeping you warm. When you wear mittens, your fingers also touch. But gloves separate your fingers. Macro: think of being in a sleeping bag with a friend or being in sleeping bags next to one another: sharing is warmer.

5. Wait 1.5 hours after sun-up. I don't know why this works, but it is substantially warmer 1.5 hours after sun-up than 1 hour after. Just do it, and thank me.

6. Run sunny roads when it's not windy and run trail when its windy. Darkness and wind are cold. Avoid when possible. However, wind is colder than darkness, I think, so if both are an issue, get in the trails. Also, running by lakes, buildings, and bodies of water is good: they radiate heat or something; it is always warmer running by a river in the winter than on a farm road.

7. Out-and-backs: Good and Bad. If you think you are up for it, an out-and-back will guarentee that you'll get in your miles. On the other hand, if it's just too cold, you are going to walk back, and that can be dangerous. Choose your routes wisely, and always have "take out points" along the way where you can bail if you feel awful.

8. When you start to retch because of cold, call it quits. Seriously. If your body's shunting blood from the stomach to keep you warm, that's a sign that it's too cold for the length of run you are doing. You don't need to die of exposure to get in a few extra runs.

9. Consistency is key. I've known plenty of runners who stayed in perfectly good shape running 3 miles six times per week, and plenty of runners who were out of shape running 10 miles twice a week. Length makes you mentally tough, but getting out there makes you fit.

10. Misery really does love company. We used to run "rant sundays"-- a sunday long run devoted to complaining-- in the winter when I was in college. Whatever it takes. When everyone is suffering, not everyone suffers as much.

11. Drink coffee/tea before you run, and eat a snack. I don't know why this works, either, but you will be much warmer while you run if you do this. A little warm fuel really helps.

12. Cover your neck. Zip your fleece all the way up, or wear a scarf if your chest gets hot. Guys, grow beards. Girls, wear hair in a ponytail under your hat so its against your neck.

13. Black pavement is bad. If the pavement looks black, it hasn't been newly cemented; that's ice. Don't run on it. Also, avoid pinestraw or anything next to a littered soda cup. No need to fall on your butt. Wooden bridges are also bad news.

14. Permafrost hurts your feet; it's easier to run on pavement in the winter because frozen, uneven ground= painful. Even if you can deal, you may get blisters, or you may get PF or ITBS. Not worth it.

Hopefully these will be inspirational for cold weather running.

A (logarithmic) leap of faith

I intended to relax tonight, but somehow, post-Starcraft, my mind ended up wandering back onto research. Funny how that works! Here's something random that I was mulling over: how do you know what form of constraint you should be using in a maximum entropy calculation? The standard variational principle for thermodynamics is maximum entropy: maximizing the entropy, S = ∑_i p_i ln p_i, subject to an average constraint on an experimental observable tells you what your probability distribution will look like at equilibrium. For example, a constraint on the system's average energy would be set as

⟨E⟩= ∑_i E_i p_i

where the angle brackets denote an average.

"Free energy" is defined as the sum of entropy and energy, or F = S +⟨E⟩, and to maximize the entropy subject to the constraint on the average energy, we set the total derivative dF = dS + β d⟨E⟩= 0, where β is the Lagrange multiplier for the constraint and d⟨E⟩= ∑_i E_i dp_i. The total derivative is

dF = ∑_i ( ln p_i + 1 + α + β E_i ) dp_i = 0

where α is the normalization multiplier, ∑_i p_i = 1, so that α ∑_i dp_i = 0. The resulting probability distribution is exponential:

p_i = e^-1-α e^{-β E_i} = Q^-1 e^{-β E_i}

where Q = ∑_i e^{-β E_i} is the partition function.

One assumption built into this calculation is that an average observed quantity implies a linear constraint on the underlying variable, E. For observed average energy, this is empirically true, but clearly for other quantities an experimental average would imply a nonlinear constraint on the variable: observed average momentum, P, for example, would constrain E = P² / 2m. This quadratic constraint results in a Gaussian probability distribution, rather than exponential:

p_i = Q^-1 e^{-β P_i2 / 2m}

This raises an interesting question: for an arbitrary observed quantity, how do we know, a priori, what the form of the constraint on the underlying variable should be? For a multiplicative process, a logarithmic constraint might be the most natural choice. (Ok, so that's not much of an argument. Well, it's late and let's face it, if you couldn't make half-baked arguments on your blog, there wouldn't be too many blogs around, would there?) In this case, for an observed value ⟨k⟩, the constraint would be

⟨k⟩= ∑_i ln (k_i) p_i

This results in a probability distribution which follows a power law,

p_i = Q^-1 k_i^-β

The challenge, then, is to decide what the most justifiable form is for the constraint, both in terms of curve fitting, as well as the underlying microscopic explanation. In the case of a logarithmic constraint, is there a solid argument, for example based on the principle that the constrained quantity must be extensive, that certain constraints should be logarithmic, rather than linear?

SAS vs. R... a fight for dominance in the statistical packages world

So I am enrolled in a statistics course that is pretty cool. I will admit that I do not know nearly as much about statistics as I want to know, and I especially struggle with remembering which tests want the P-value to be less than the statistic and which want it to be more in order to reject H naught (okay, well, saying that, I now remember: Shapiro Wilks and Levine want it to be more, but I forget in key situations, like, uhm, exams).

Anyway, we've just received our first "homework" and as nerdy as this sounds, I am excited about it... except for one little thing... it's done on SAS.

Now, I can't claim to be an expert in SAS or in R, and maybe someone who is will give me a little tip on why my opinion is complete fail, but I really think R is much more user friendly, especially for someone like me who has very little programming experience outside of making macros in Excel (prior to the great loss of VBA apps in Excel 2008 for Mac... the horror!), making a few visual basic programs that can taunt the user with insults given a certain input, or making annoying pop ups when certain coordinates on a screen are moused over. In other words, I know almost nothing about programming.

For example, I am about to begin a program in SAS for making a correlation matrix. Right, simple, a correlation matrix. Let me show you what that would look like in R.

DATANAME<-as.matrix(read.table('/R/dataname.csv', sep=","))
#possibly a few specifications about how to read the data included in that first bracket
DATACORR<-corr(dataname)
DATACORR

That's it!! That's the whole correlation matrix for R. It will show up immediately after you type in that second DATACORR (R likes to print automatically, oh, how Ilove R!)

Now let's talk about how we do THE EXACT SAME THING in SAS

DATA DATANAME;
INPUT VAR1 VAR2....etc. #not going to input vars here- worthless for the rant
# here is where you input all of the relationships between the variables if you need to
DATALINES;
#now you type in all the data, because SAS really hates reading from files... or you can be lazy like me and use Excel's export features to make files that are suitable for SAS... and easily adaptable to the variables you want to input
PROC CORR NOSIMPLE DATA= DATANAME;
VAR VAR1 VAR2 VAR3 .... etc.
WITH VAR1 VAR2 VAR3 ... etc.
RUN; QUIT;

Now, I think we can all see the problem here; as your data set gets bigger and bigger, and it will, incorporating all this stuff into SAS... pain in the butt. For SAS experts, there may be some shortcut to doing this, but let's think about it for a second. I am not a SAS expert, and many people who need to use statistical packages for large data set applications (biologists, etc.) probably aren't programming experts, either... So this seems like a lot of trouble to me.

With that being said, I think a fight needs to go down: SAS v. R... who will dominate the "Entry-Level Statistical Packages World"? I am obviously biased to R. Here's some points to think of

FOR SAS:
- more traditional package, has been around for a long time
- already incorporated into curriculums
- produces standard plots
- labels statistical tests and their solutions (for example, on the SW test it will remind you that PW) clearly
- can easily run "tests" for equality of B's that help choose model form
- traditional ("old-school") programming syntax

FOR R:
- really sweet graphics
- lots of cool packages to download
- easy commands to use
- reading files is not TOO difficult (it can even work with some Excel files-- which is amazing-- it knows the data entry from the function entry!)
- calculates statistical tests
- new-school syntax
- 3D graphics that look just amazingly sweet (2 comments on the graphics, I know, but you can make just about anything on there! You can layer it with a MAP for goodness sake... I mean... hot skippy, that's amazing!)

So... it's the duel of the century in nerd world. I guess as this class progresses we will see-- will my exposure to SAS build my love for it or will my heart remain devoted to R? More of this epic saga to come.

Mass Carriers and Britain from the Sky

Well, it appears that I am now a bunch of mass carriers... :)

Nothing wrong with that! In a world of computerized battling, I dominate. That being said, this is a world in which hordes of flying centipedes often randomly fly around, shooting green vomit out of their mouths, and armored tanks engage in battles with giant, glowing white men who can cast spells.

In other words, it is exactly like real life.

Well, carriers aside, I am here more to post another useless link that should keep you from doing work for at least the next few minutes:

http://www.dailymail.co.uk/news/article-1197898/Pictured-The-Google-Earth-alphabet-Britains-hedgerows-roads-buildings.html

You may think, well, that's not all that impressive, but one thing I have learned from working with remote sensing is... it's pretty hard to do! You've got to get those coordinates just right. In fact, one of the coolest things I have on my desk right now is a copy of something a friend of mine here at Clemson made for GIS class-- it's a similar image to the Britain one above, but instead it has "Clemson University."

So that impressed me for the day; I've heard also that Britain is currently "iced over" and that this can also be seen on Google earth. I've been meaning to check this out. Speaking of ice, I had a pretty stellar run today; it was about 24 degrees here and I didn't think that the ice storm had actually affected Clemson, but part of my run was through the back of the forest by the dikes, and there were several trees that were wearing ice jackets. When I looked up over the golf course, some of the lower Appalachians were snow-dusted. One of my favorite things about this place is how well the people know these foothills. I mentioned this afternoon to someone that I had seen the snowdusting; they asked where; I replied "on the fourth mountain over from table rock, and also the long flat one with two peaks." Immediately my respondent knew exactly what I was talking about. I thought during today's run, you know, Clemson isn't the most exciting place in the world, but its wonderfully authentic. The people who live here know this land in such an intrinsic way, and as much deer as they shoot and beer as they drink, I think there's something to be said for that.

I feel also in light of the brilliant quotes included in previous posts by Pericles I should therefore also add in a particularly awesome quote... and may I be one of the few Christian folk out there who thinks that Nietzsche, yes, Nietzsche, says one of the most beautiful lines I have ever heard:

"I still live, I still think: I still have to live, for I still have to think. Sum, ergo cogito; cogito, ergo sum. Today everybody permits himself the expression of his wish and his dearest thought; hence, I, too, shall say what it is that I wish from myself today, and what was the first thought to run across my ehart this year-- what tought shall be for me the reason, the reason, warranty, and sweetness of my life henceforth. I want to learn more and more to see as beautiful what is necessary in things; then I shall be one of those who makes things beautiful. Amor fati: let that be my love henceforth! I do not want to wage war against what is ugly. I do not want to accuse. Looking away shall be my only negation. And all and all and on the whole, someday I wish to only be a yes-sayer."

Okay, well I hope you enjoyed, and at least maybe if you hadn't heard that Nietzsche quote you thought, well, that's an interesting quote.

By the way, Rice Chex are now gluten free, and that's a big win.

When Street Luge is just not hardcore enough

So, the other day I was curious-- what are the most extreme sports out there?-- My list, prior to being informed, would have been something like this:

1. Ultramarathoning

2. Kite-boarding

3. Elk hunting

4. Rugby (I mean, you've seen those dudes after a game, that's frickin' hardcore)

5. BJJ

6. White Water Kayaking

7. Mountain Boarding

8. Trick Bikes (like on the X-games, those bikes that do tricks similar to skateboards/roller skates)

9. Free climbing

10. Free running (parkour, I think, is the formal name)

11. Luge (in all forms... street, snow, ice, fire, whatever)....

12. "Ghost Ride the Whip" (may not actually be a sport but... come on, you've gotta love some ghost riding... for those of you who don't know what ghost riding is, please youtube it... hours of entertainment await)

13. Being an astronaut (not sure if that's a "sport" but I'd do it for recreation!)

14. Bowfishing (Hardcore points are owed to bowfishers-- They are shooting a FISH with a BOW AND ARROW!!-- the physics of doing that alone boggle me, not to mention the fact that you have to be able to shoot a bow and arrow extremely accurately)

15. Speed skating

Anyway, those seemed pretty awesome to me... until I looked on my good friend, the World Wide Web, and found out that I have a really sissy perspective on the term "hardcore."

Well, I found this:

http://www.oddee.com/item_96668.aspx

Okay, so not all of it is that special, but check out sport number 2: Limbo Skating... I mean... that's ridiculous. Everything in my life that I thought was hardcore has just become equivalent to eating a pile of marshmallows.

I'm going to go make mass carriers now so that I feel like a bad @$$ again.

Thing that make the world implode

Decaf americano?

I've heard this ordered at Starbucks(es) before... decaf americano. For those who do not know, americano is coffee language for espresso and water... aka "watered down espresso" for those who are too pansy to drink the real thing... or people like me who want the satisfaction of a whole mug without sacrificing the caffiene. But a decaf americano? What is the point of that... "Yes, hello, I would like watered-down fake coffee." FAIL.

The "luxury utility vehicle" (LUV)?

This includes all "SUVs" that are not made for "S"-- Acura MDX, Porsche Cayenne, etc. What is the point? Who buys a car for utility when it is not going to be utilized for anything more than driving around shopping. Shopping is not a sport. And SUVs should not have seat warmers. I think these cars exist to show that the consumer has the money to guzzle gas without hesitation. Either do something sporty... or don't. It's like the car version of "is cheerleading a sport?"

The Hannah Montana Guitar?

Is it Miley Cyrus? Or is is Hannah Montana? Is it a guitar... or isn't it? World ----> explode.

Snuggie?

Because it's cool to wear your blanket out of the house. If you are too lazy to take your arms from out from under the blanket to get the remote, but can still venture to walmart to stock up on all the latest consumer goods... while wearing your snuggie? Uhm. Yeah. And now they have those fashion snuggies? Making wearing blankets in public-- stylish?

The Elliptical Trainer?

So the idea is that you put your feet into these holsters that move your legs for you in a circular motion and somehow you burn more calories than running? How does that work? All I am going to say is... I have seen many fit athletes who run. I have not seen anyone who is a champion of ellipticalling. "Oh, look at me, I'm the BEST ELLIPTICALER IN THE WORLD. I can burn 100000 calories in an hour!" No.

Fur lined boots and shorts/skirts?

Why.... (EXPLODE)

Warranties on Running Shoes?

For ten dollars, you could protect your shoes against wear and tear. But running is not included in the list of actions protected against. I don't know about you, but the next time I go ice climbing in my DS Trainers I am taking that action back to sportsauthority. Unusual wear and tear? Think you got it backwards there, SA.

Jolly Ranchers and Bacon.

There is at least one thing in the world that does NOT go with bacon.

Mathematics

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
- Bertrand Russell

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
- Henri Poincare

"Similarly, mathematics can achieve a proof that has no interaction with the physical world. It may even be the closest to divine truth that human beings can achieve."
- Andrew Sullivan

Solitude

"He who delights in solitude is either a wild beast or a god."
- Francis Bacon

"I lived in solitude in the country and noticed how the monotony of a quiet life stimulates the creative mind."
- Albert Einstein

"Language... has created the word "loneliness" to express the pain of being alone. And it has created the word "solitude" to express the glory of being alone."
- Paul Tillich

Politics

"The word 'bipartisan' usually means that some larger-than-usual deception is being carried out."
- George Carlin

Meditation

"The birds have vanished in the sky, and now the last cloud drains away. We sit together, the mountain and I, until only the mountain remains."
- Li Po

Effort

"It is not the critic who counts: not the man who points out how the strong man stumbles or where the doer of deeds could have done better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood, who errs and comes up short again and again, because there is no effort without error or shortcoming, but who knows the great enthusiasms, the great devotions, who spends himself for a worthy cause; who, at the best, knows, in the end, the triumph of high achievement, and who, at the worst, if he fails, at least he fails while daring greatly, so that his place shall never be with those cold and timid souls who know neither victory nor defeat."
- Theodore Roosevelt

Freedom

"Live free or die. Death is not the worst of evils."
- John Stark

Commencement

"This, I submit, is the freedom of a real education, of learning how to be well-adjusted. You get to consciously decide what has meaning and what doesn't. You get to decide what to worship.

"Because here's something else that's weird but true: in the day-to-day trenches of adult life, there is actually no such thing as atheism. There is no such thing as not worshipping. Everybody worships. The only choice we get is what to worship. And the compelling reason for maybe choosing some sort of god or spiritual-type thing to worship--be it JC or Allah, be it YHWH or the Wiccan Mother Goddess, or the Four Noble Truths, or some inviolable set of ethical principles--is that pretty much anything else you worship will eat you alive. If you worship money and things, if they are where you tap real meaning in life, then you will never have enough, never feel you have enough. It's the truth. Worship your body and beauty and sexual allure and you will always feel ugly. And when time and age start showing, you will die a million deaths before they finally grieve you. On one level, we all know this stuff already. It's been codified as myths, proverbs, clichés, epigrams, parables; the skeleton of every great story. The whole trick is keeping the truth up front in daily consciousness.

"Worship power, you will end up feeling weak and afraid, and you will need ever more power over others to numb you to your own fear. Worship your intellect, being seen as smart, you will end up feeling stupid, a fraud, always on the verge of being found out. But the insidious thing about these forms of worship is not that they're evil or sinful, it's that they're unconscious. They are default settings.

"They're the kind of worship you just gradually slip into, day after day, getting more and more selective about what you see and how you measure value without ever being fully aware that that's what you're doing.

"And the so-called real world will not discourage you from operating on your default settings, because the so-called real world of men and money and power hums merrily along in a pool of fear and anger and frustration and craving and worship of self. Our own present culture has harnessed these forces in ways that have yielded extraordinary wealth and comfort and personal freedom. The freedom all to be lords of our tiny skull-sized kingdoms, alone at the centre of all creation. This kind of freedom has much to recommend it. But of course there are all different kinds of freedom, and the kind that is most precious you will not hear much talk about much in the great outside world of wanting and achieving.... The really important kind of freedom involves attention and awareness and discipline, and being able truly to care about other people and to sacrifice for them over and over in myriad petty, unsexy ways every day.

"That is real freedom. That is being educated, and understanding how to think. The alternative is unconsciousness, the default setting, the rat race, the constant gnawing sense of having had, and lost, some infinite thing."

- David Foster Wallace

The long way home

It was hot today! I'd forgotten how much I miss the heat. Even at 8:30 this morning it was too warm for gloves on my motorcycle, and it got up into the 80s as the day went on. It felt wonderful.

I got off work at about 8, and it was still warm and breezy. Instead of going up onto the highway, on an impulse, I drove west through the city, looking for a slow route home. I meandered through the colorful, off-beat Mission District until I happened on a hole-in-the-wall sushi place called Yo's Sushi Club. It felt authentic, with a clean, very Japanese aesthetic. The chef, Yo, stood behind the counter, a white towel wrapped tight around his bald head. There was no bento or teriyaki on the menu, just sushi, and I watched as Yo made all the sushi himself. The chirashi was delicious. I ate slowly, then rode the rest of the way down Mission until I found myself on the hilltop at Daly City. It was starting to cool off by the time I made my way to the road across the mountain, and saw the lights of San Francisco spread out north of me as I wound through the pass.

I've got to take the long way home more often. I hope the weather stays nice! If it's clear, tomorrow night I'm going with some friends out past Livermore in the east bay, and we're going to watch the meteor shower.

If I had a million dollars...

I worked a half day today. I didn't have a particular reason for going in, but there were a few things I wanted to catch up on, so I fired up the bike and rode up to campus. I spent most of the day sitting at my desk, reading a mathematical physics textbook, and tinkering with a simulation I'd written earlier in the week.

A recurring theme of me-at-work, pretty much ever since I started working when I was 15, is the thought, Sigh...if only I had a few million dollars, so I could do X instead of this... Today, it struck me that, if I was independently wealthy and could do whatever I wanted, I probably would spend my time doing what I did today. I'd find somewhere quiet to sit, and learn some new physics.

On top of that, I found out a couple of weeks ago that I won a competition for a federal research fellowship, so I'll get an extra $4000 or so per year to do exactly what I'd be doing anyway, with no strings attached. (I worked out the total value of the fellowship, and was surprised at how large it is: it pays for everything for three years, which comes out to be around $150,000!) As an additional bonus, it's considered very prestigious, looks good on a CV, etc. (not that I'll be applying for jobs anytime soon...).

I hesitate to say so, but I feel honestly satisfied with the way things are going. It's the first time I've felt this way in a long time, so it's a bit alien to me, but...I'm happy. (On a related note, I also feel completely vindicated in my decision to switch research groups.)

Everyone loves grad school!

I dug up a funny transcript of a conversation I had with my brother a few months back...pretty much captures how much I hated my previous lab:

2:49:47 PM Gheed: man grad school blows though
2:49:50 PM Me: lol
2:50:00 PM Me: WUT DO U MEAN IT'S AWESOME
2:50:01 PM Me: :\
2:50:51 PM Gheed: >:o
2:51:08 PM Me: >:o indeed
2:51:22 PM Gheed: u have any idea how useful a "masters of science" degree is?
2:51:29 PM Me: i wake up every morning and think to myself: >:o
2:51:47 PM Me: i do not
2:51:51 PM Me: inform me immediately
2:52:23 PM Me: btw i do a lot of >:o while i'm at work too
2:52:31 PM Me: then when i get home i'm all like >:o
2:52:45 PM Gheed: lol yeah that face sums me up pretty well too
2:52:47 PM Gheed: >:o
2:53:04 PM Me: there's a few :\ and :( thrown in there for good measure, too
2:53:12 PM Me: but for the most part it's all about the >:o
2:53:28 PM Gheed: >:o
2:53:32 PM Me: >:o

Counting things

I've got a cup that has a little glass wall dividing it in half. It's half full of salt water, and half full of fresh water. If the glass wall is suddenly removed, then the salt water mixes with the fresh water, until the salt ions are evenly distributed in the cup. This is diffusion: the tendency of particles to move from regions of high concentration to regions of low concentration.

A simple way of thinking about diffusion is to imagine the cup is composed of a large set of very small blocks, big enough to hold a salt ion. Each block may be filled or empty. (This is called a lattice model.) If there is no bias built in to the particles, the probability of each system arrangement should be equal. An even distribution of particles is the most probable system configuration because it contains the highest number of total possible arrangements.

To see this, imagine there are four salt particles in a very small cup, and we partition the cup into four little boxes in the left half, and four more in the right. How many ways are there to have all four particles on the left side, and none on the right? Assuming the particles are indistinguishable (that is, we don't care what order they're in), since there are only four boxes, there is only 1 system configuration that has all four particles on the left side. On the other hand, there are 6 different ways to arrange the particles to have 2 on the left side and 2 on the right. If each individual arrangement is equally likely, the most probable state in which to find the system is with 2 particles on the left, and 2 on the right. To be precise, this would be six times more likely than having all the salt on one side! (Mathematically, these are just binomial coefficients, so if you're lazy like me and don't like doing a bunch of counting, you can get the number of arrangements by reading across the fifth row of Pascal's triangle, 1 4 6 4 1.)

The idea of maximizing the number of states is central to statistical mechanics. The number of states (or, more specifically, its natural logarithm) is a measure of the amount of disorder, or entropy, in a system. One of most important conclusions of thermodynamics is that in a system with a conserved amount of total energy (and number of particles, taking the classical view that these are distinct items), its entropy will always increase. (This has a number of interesting, and disquieting, implications for, for example, the future of our universe, if it is an isolated system.) This counting approach is a microscopic model for systems at equilibrium, but in a real system -- e.g., a real-life cup of water -- of course we don't sit there and count the number of salt ions in the cup. In one gram of table salt, there's around 3.6x10²⁵ pairs of sodium and chloride ions. This is a ridiculously huge number: it's 36 followed by 24 zeros, or 36 trillion trillion ions in a single gram. If you consider the number of arrangements of 36 trillion trillion ions where all the ions are on one side of the cup (total: 1), versus the number of arrangements where they are approximately equally distributed (total: a stupidly big number), it's clear that in this simple lattice model, generally all the salt is not going to be on one side of the cup!

Instead of counting trillions of ions, instead what we measure are macroscopic quantities -- for example, the average concentration of ions on each side of the cup. If we take this measurement, and discover that there's actually a higher concentration on the right side than the left on average, then that tells us something about our system: it gives us a constraint that we can use to weight the entropy calculation we did before. As mentioned before, if our system contains 36 trillion trillion particles, we're probably only going to be interested in average quantities, since the stupidly big number-of-states will make shifts away from the average very improbable: the probability distribution, in this case, will be approximately Gaussian (a bell curve), with a tiny standard deviation. But, what about our oversimplified tiny cup with only four particles of salt in it? In that case, there's a 1 in 6 chance of finding all the salt on one side of the cup! Given this non-negligibly small probability, for very small systems, the microscopic fluctuations can be important. That is to say, we're interested in how the system evolves with time -- we care about its dynamics.

Let's go back to our tiny cup, and start with 2 particles on the left side, and 2 on the right. Assume there's no bias built in to the system, and that every particle is independent of the others, so that each particle has some fixed probability of jumping to the other side. The sequence of jumps and stays that a particle follows over a period of time is called its trajectory. If there are no constraints on the system, we'd predict that every trajectory is equally likely, and if we tabulate the number of trajectories that result in 0, 1, or 2 particles jumping, 6 out of 16 trajectories result in 2 particles on the left, and 2 on the right (the same result as before). However, the table also says that there is a 1 in 16 chance for 2 particles to jump from left to right, and 0 to jump from right to left, putting all the salt on one side of the cup! For a given starting distribution of particles, this quantifies how the system's fluctuations are likely to alter the overall state of the system with time, and also how the fluctuations themselves are likely to change over time. A system's trajectory multiplicity is called its caliber.

We think that the idea of maximum caliber will be a useful way to analyze systems far from equilibrium, very small systems, as well as highly correlated systems. All three are true of many biological systems, and one system I'm particularly interested in applying maximum caliber to is long term potentiation, the way we form memories. My short-term project is a little farther afield: I'd like to apply caliber to correlated stock prices, and see if the predicted fluctuations in the stock market correspond to what's actually been observed. Because stock prices can be highly correlated, the independent-particle assumption does not work, which is why random-walk statistics do not accurately describe the probability of large fluctuations in the market. Instead, markets approximately follow an inverse quartic power law, but as far as I'm aware, no one's provided a quantitative, microscopic explanation for this. It will be interesting to see if a power law distribution follows naturally from thinking about the caliber of the market.

An anecdote

"I'd like to spend a week or two doing some reading," I said to my new research advisor, vaguely apprehensive. "I want to get a better feel for what's been done and how master equations work, and to review some math and physics I don't remember as well as I'd like."

He grinned at me. "That's fine. Work at your own pace."

I returned the grin. Suddenly, the future looks very bright.