This code implements the stochastic Markov model of Astumian [D.R. Astumian, Am. J. Phys. 2005, 73(2):178-183] and it's (seemingly) paradoxical behaviour. Games labelled as 0 or 1 are both both probalistically losing, yet randomly switching between the two games (denoted Game 2) yields a winning outcome on average. The initial condition is the middle state, i=2 (phi=0.5) and transitions proceed until either losing (0/phi=0) or winning (4/phi=1) states are reached.
An interesting addition to this story is that using a correlated/biased fluctuation spectrum (denoted Game 3) results in net losing in excess of simply playing a single game!
We describe each state point by an order parameter, which is either 0, 0.25, 0.5, 0.75 or 1. When drawing a random number to decide which game to play, rather than draw a RN from a uniform distribution, [0,1], we draw from [0,phi]
The results of this toy model highlight how non-equilibrium systems, subject to thermal noise (uncorrelated or correlated) driving a system across a rough energy landscape, can drive reactions in reverse.
@2009, Jonathan Belof
After obtaining the source code, please consult the Makefile to set any specific compiler flags (defaults are gcc with std C library).
The code implements direct numerical simulation of a Markov Chain Monte Carlo for two different reaction networks, with specific barriers and energy wells, such that "playing" either one of them leads to a "losing" game on average. Randomly alternating between games leads to a "winning" result.
Compilation is simple and relies on only standard libraries:
$ make gcc -c -O3 -DDEBUG -I. astumian_game.c gcc -O3 -DDEBUG *.o -o ag
Run the binary without arguments to obtain the usage of command line input:
$ ./ag ./ag: - integer game-type of 0,1, 2 (uniform) or 3 (correlated) - integer number of steps to perform
To run the (losing) Game 0:
$ ./ag 0 1000000 Running game-type 0 Running 1000000 simulation steps losses / wins = 555560/444440 = 1.250023 analytic: game0 = 1.250000, game1 = 1.250000, uniform = 0.810000
where the MCMC results are shown in the penultimate line, compared with the clsed form analytic result obtain via product of state probabilities.
Likewise for running Game 1:
$ ./ag 1 1000000 Running game-type 1 Running 1000000 simulation steps losses / wins = 555588/444412 = 1.250164 analytic: game0 = 1.250000, game1 = 1.250000, uniform = 0.810000
with the explicit numerical calculation approaching the analytic result with more instantiations.
With Game 2, we flip a coin and randomly select Game 1 or Game 2 Markov probabilities at each transition point; playing this hybride game leads to a reversal of the system!
$ ./ag 2 1000000 Running game-type 2 Running 1000000 simulation steps losses / wins = 446432/553568 = 0.806463 analytic: game0 = 1.250000, game1 = 1.250000, uniform = 0.810000
(note that the analytic result to compare with is the 3rd one called "uniform").
- Jon Belof jbelof@github
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This project is licensed under the GNU General Public License v3, please see GPL_license.txt for details.