Concise model of modern fiat money and its non-conservation A confession:  I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply.  By the standards of someone trained in mathematics, all of the explanations that I have ever seen are either too short or too long.  My impression is that the way that a central bank controls the money supply in a modern economy can be taken on faith (if you want a short explanation), or is hard to understand (if you want a long one), but I have am suspicious of both of these extremes.  I have also seen explanations that describe what happens "in effect" without clearly explaining the underlying rules.  I would be interested in a concise mathematical summary of how a currency such as the US dollar is controlled.  (I hope that it can be taken as an MO-appropriate question in mathematical economics.)
Here is a model that I understand, but that isn't true:  A game such as Monopoly has a central bank that simply grants fiat money from time to time to private parties.  I'm sure that this is the wrong way to run a real economy, but at the serious level I don't know why.  In any case this is not how the Fed works, because it mostly lends money rather than simply granting it.
Here is a failed improvement of the model:  Suppose that the bank in Monopoly only lent money to the players instead of granting it.  Then the players would have no way to pay back the loans with interest!  Maybe it could work if the players were allowed to accumulate debt --- but what would prevent unlimited borrowing?
I can believe in multiplier effects (although actually I don't know a rigorous definition).  If transactions occur more and more quickly, or if assets get more and more leveraged, that could be equivalent to an increase in money.  I have trouble believing that the central bank does not need to create money and that we see inflation (except in depression circumstances) solely because money keeps travelling faster and faster and because the economy gets more and more leveraged.
An abstracted economy has the following actors, each operating according to certain financial rules: A central bank, a government budget, regulated private banks, and the rest of the private sector.  (And foreign actors, who I suppose are an extension of the private sector.)  I think that I know the basic financial rules for the last one, but not for the others.  To rephrase the question, I am hoping that there is a concise mathematical model that makes clear when money is created, and that looks dynamically stable with some controllable rate of inflation.  A reference could be okay, but only if it has a good, specific explanation.
 A: Thanks to the comments and answers from Scott Carnahan and Michael Greinecker, I think that I understand it better now.  I'm going to write this as a CW summary answer and also accept one of the other answers.
People often talk as if all currency is borrowed from the central bank, but that is not really true.  If it were literally true, it wouldn't make sense, because there would be no way to pay the principal and interest back to the central bank.  Or otherwise, if all money is borrowed, then an economy's total monetary assets stay at zero, which is not strictly impossible, but doesn't sound right.
What I guess actually happens is that the central bank both buys and sells treasury bonds.  Even though this is done for interest rate stability, the central bank is perfectly happy to sell high and buy low, thereby violating conservation of money.  It is also counterintuitive in the following respect:  Although in the short term a high interest rate contracts the money supply, in the long term the interest paid expands it again.  Nonetheless, I guess that the demand to have money to trade sustains the value of the money and keeps everyone from just buying treasury bonds at high interest.  I guess here you would point to the money supply equation that Steven Landsburg posted.  (It does not leap out at me that it really leads to currency stability, but I can believe it.)
Also, to get a currency started, the central bank can first buy or sell other commodities, for instance gold, so that the private sector then has money to buy treasury bonds.  Another counterintuitive point (but one that doesn't bother me) is that if the central bank trades commodities at a monetary "loss", then actually it has gained those commodities.  This inverted mode of gain by a bank seems to be one meaning of "seigniorage".  Another meaning is any increase of the money supply from the central bank's trades, so at some level seigniorage is the main answer to my question.

Another player is the national government.  Unlike the private sector, it is allowed an unlimited amount of debt.  So, a second non-conservation of money is deficit spending, if in tandem the central bank keeps lowering the interest rate.  Unlike seigniorage, this may be de facto non-conservation of money, but it is not de jure non-conservation of money, if the government keeps an honest account of how much it borrows.  (As Deane and Michael discuss, this honesty is only really possible if the central bank is politically independent from the government budget.)
A third type of non-conservation of money is a default by a commercial bank that owes money to the central bank.  But this does not look like a natural way to increase the money supply, and I don't think that it is.
A: I feel like I'm missing something dumb but it seems to me that fractional-reserve banking is not referenced or explained in the question or any of the answers.  People here DO understand it, right?  It is where money actually comes from.  Basically if you borrow \$1000 from a bank, they can issue the loan by typing something into a computer that increases your account balance by \$1000 without subtracting from some other account.  The \$1000 doesn't have to be transferred from anywhere, but rather it is created out of nothing, which is why money is not a conserved quantity.  There are of course a bunch of constraints such as the reserve requirement, but money created by banks through lending (the regulatory ability to create money that way is what distinguishes a bank from, say, a payday lender that has to transfer money from itself to you instead of creating it) is the cause of the non-conservation.
http://en.wikipedia.org/wiki/Fractional_reserve_banking
A: I think an answer that discusses the actual institutional details of how the Fed controls the money supply would be off-topic here. Also, the Fed works slighlty differently from the ECB in that regard and there is more than one method of influencing the money supply (take a look at the wikipedia page on money creation). So I will try in this answer to demystify how a central bank can create money without literally sending out helicopters that drop fiat money on people.
First, one has to get right what money is. In explicit formal models, money is an asset that never pays out. If it has value, it is because there is a bubble in this asset. The first such model of money can probably be found in the 1958 paper An Exact Consumption-Loan Model of Interest with or without the Social
Contrivance of Money by Paul Samuelson. It is worth pointing out that bubbles are not inherently bad and that paper constructs a toy economy in which everyone profits from the money bubble. 
Now how can one increase the supply of an asset that never has to pay out anyways? It sells the asset in exchange for other assets. Since money never has to pay out, the central bank will not face a solvency constraint in the process. Selling money is not that different from selling milk, but since there are no cows involved, central banks are not constrained by cost.
A: Turning my last comment into an answer:
The simplest model of money demand is $M=M(P,Y,i)$ where $P$ is the price level (if all prices rise, you'll probably want more money in your pocket), $Y$  is real income (if you're richer, you might want more money in your pocket) and $i$ is the nominal interest rate (if the interest rate rises, you'll want to hold more bonds and consequently less money).  
In the simplest models, $Y$ is determined by non-monetary factors, and (thinking now of everything as a function of time) $i=r+P'(t)$ (where $r$ is determined by non-monetary factors).  This follows from the assumption that prices are perfectly flexible, so that $Y$ has to be determined by supply and demand in the markets for goods and labor.
At time $t$, the money supply is $M_0(t)$, where $M_0$ is a function chosen (in the simplest models) by the Fed.  Equilibrium requires $M=M_0$.  (If, for example, $M$ is less than $M_0$, so that people are unwilling to hold $M_0$ dollars, they will attempt to dispose of dollars by exchanging them for goods, which bids up $P$ and causes $M$ to rise.  Likewise in the opposite direction).  
So the key equation is $M(P(t),Y(t),r+P'(t))=M_0(t)$ with $Y(t)$, $r$ and $M_0$ determined outside the model.  
A more sophisticated model would make $M(t)$ dependent on expected future values of $P$ and $i$, and include an account of how those expectations are formed.  So you should view this as the freshman version of the story, not the grad school version.
A: For a mathematical model see Hayashi and Matsui, 1994. For an in-depth discussion without too many (actually, any) equations, see many books by Murray Rothbard (all available on Amazon.com).
A: Although not a mathematical concise definition but empirical evidence that money is created "out of thin air"..
"It was examined whether in the process of making money available to the borrower the bank transfers these funds from other accounts (within or outside the bank). In the process of making loaned money available in the borrower's bank account, it was found that the bank did not transfer the money away from other internal or external accounts, resulting in a rejection of both the fractional reserve theory and the financial intermediation theory. Instead, it was found that the bank newly ‘invented’ the funds by crediting the borrower's account with a deposit, although no such deposit had taken place. This is in line with the claims of the credit creation theory."
"Thus it can now be said with confidence for the first time – possibly in the 5000 years' history of banking - that it has been empirically demonstrated that each individual bank creates credit and money out of nothing, when it extends what is called a ‘bank loan’. The bank does not loan any existing money, but instead creates new money. The money supply is created as ‘fairy dust’ produced by the banks out of thin air."
http://www.sciencedirect.com/science/article/pii/S1057521914001070
A: I think economics is far more closely connected with the body politic than it with mathematics. Also applying mathematics to economics is also a political act and a signification. (Mathematics has an association with permanance which can be used symbolically to shore up a certain contingent political/economic order).
Physics examines the world by supposing the physical world follows a rational order, and that by dint of effort this order is discoverable. I can't see how this applies to the social order of societie(s); how does one measure wealth, imagination, violence, ethics, power, desire, criminality?   
Whereas mathematics applied to physics captures something of its fundamental relationships, it appears to me that a mathematical model of a social order can only captures superficial and contingent things. 
