THE LANDAUER LIMIT AND THERMODYNAMICS OF BIOLOGICAL SYSTEMS

Transcription

1 THE LANDAUER LIMIT AND THERMODYNAMICS OF BIOLOGICAL SYSTEMS Daid H. Wolpert Santa Fe Institute, MIT ttp://daidwolpert.weebly.com Jos Grocow Santa Fe Institute

2 HEAT OF ERASING A BIT a) b) c) R 1 2 2R 1 2 R 1 2 Termodynamic cost to erase a bit - te minimal amount of entropy tat must be expelled to te enironment - is ln[2]

3 HEAT OF ERASING A BIT a) b) c) R 1 2 2R 1 2 R 1 2 Crucially, DO know precise pre-erasure alue of bit - After all, a computer is useless if don t know its initial state Bennett, 2003: If erasure is applied to random data, te operation may be termodynamically reersible but if it is applied to known data, it is termodynamically irreersible.

4 HEAT OF ERASING A BIT a) b) c) R 1 2 2R 1 2 R 1 2 Crucially, DO know precise pre-erasure alue of bit - After all, a computer is useless if don t know its initial state In fact, te prior distribution oer te pre-erasure alue and in particular te entropy of tat prior is irreleant Requires careful engineering to make tis property old Local detailed balance does not old

5 REFRIGERATION BY RANDOMIZING A BIT A) B) C) R 1 2 R/2 1 2 R 1 2 Example: Adiabatic demagnetization Exploited in modern engineering: o Noisy error correction computing o Real (not pseudo ) random number generators

6 SOME ERASING AND SOME RANDOMIZATION Wat is te termodynamic cost for an arbitrary conditional distribution from X = {0, 1, 2, 3} into itself? E.g., wat if 0 and 1 go to 0 (as in bit erasure); i.e., P(0 0) = P(0 1) = 1 2 goes to 0 wit probability.8, stays te same oterwise; i.e., P(0 2) =.8, P(2 2) =.2 3 goes to 2 wit probability.4, and to 0 wit probability.6; i.e., P(2 3) =.5, P(0 3) =.6

7 THERMODYNAMIC COST E(cost) = X apple X ( t+1 t )P ( t )ln t, t+1 = I(W t+1 ; V t+1 ) I(W t ; V t ) 0 t ( t+1 t ) were t is te obserable s alue at time t; π(..) is te conditional distribution of dynamics; I(. ;.) is mutual information; W is unobsered degrees of freedom;

8 THERMODYNAMIC COST E(cost) = X t, t+1 ( t+1 t )P ( t )ln apple X 0 t ( t+1 t ) were t is te obserable s alue at time t; π(..) is te conditional distribution of dynamics; Example: In a 2-to-1 map, π(0 0) = π(0 1) = 1, so expected cost equals ln[2]

9 BOUNDS ON THERMODYNAMIC COST Gien te eolution kernel π(..), as one aries P( t ): 0 apple E(cost)+H(V t )+H(V t+1 ) apple log[ V ] max t were t is te obserable s alue at time t; H(.) is Sannon entropy; KL(..) is KL diergence; Π + ( t+1 ) = t π ( t+1 t ) / V apple KL (V t+1 a) + (V t+1 )

10 BOUNDS ON THERMODYNAMIC COST Gien te eolution kernel π(..), as one aries P( t ): 0 apple E(cost)+H(V t )+H(V t+1 ) apple log[ V ] max t were t is te obserable s alue at time t; H(.) is Sannon entropy; KL(..) is KL diergence; Π + ( t+1 ) = t π ( t+1 t ) / V apple KL (V t+1 a) + (V t+1 ) Example: In a 2-to-1 map bot bounds are tigt: Termodynamic cost is te drop in Sannon entropies oer V

11 K TH ORDER MARKOV CHAINS For a k t order Marko cain, termodynamic cost during a single step is bounded below by L H(V t V t+1,...,v t+k 1 ) H(V t+k V t+1,...,v t+k 1 ) and aboe by Very messy expression

12 K TH ORDER MARKOV CHAINS For a k t order Marko cain, termodynamic cost during a single step is bounded below by L H(V t V t+1,...,v t+k 1 ) H(V t+k V t+1,...,v t+k 1 ) and aboe by Very messy expression If P() as reaced stationarity, lower bound is E(cost) = I(V t ; V t+1,...,v t+k 1 ) I(V t+k ; V t+1,...,v t+k 1 ) (Cf. Still et al. 2012)

13 THERMODYNAMIC COST Tons of oter fun results, including: 1) Second law: Termodynamic cost is non-negatie for any process tat goes from distribution A to distribution B and ten back to distribution B 2) Examples of coarse-graining (in te statistical pysics sense) tat increase termodynamic cost 3) Examples of coarse-graining tat decrease termo. cost 4) Implications for optimal compiler design 5) Analysis of termodynamic cost of Hidden Marko Models

14 IMPLICATIONS FOR DESIGN OF BRAINS P(x t ) a dynamic process outside of a brain; Natural selection faors brains tat: (generate t s tat) predict future of x accurately; but not generate eat tat needs to be dissipated; not require free energy from enironment (need to create all tat eat) Natural selection faors brains tat: 1) Accurately predict future (quantified wit a fitness function); 2) Using a prediction program wit minimal termo. cost

15 IMPLICATIONS FOR BIOCHEMISTRY Natural selection faors (penotypes of) a prokaryote tat: (generate t s tat) maximize fitness; but not generate eat tat needs to be dissipated; not require free energy from enironment (need to create all tat eat) Natural selection faors prokaryotes tat: 1) Beae as well as possible (quantified wit a fitness function); 2) Wile implementing beaior wit minimal termo. cost

16 COMPLEXITY DYNAMICS OF BIOSPHERE E(cost) = X apple X ( t+1 t )P ( t )ln ( t+1 t ) t, t+1 were t is te obserable s alue at time t; π(..) is te conditional distribution of dynamics; 0 t N.b., termodynamic cost aries wit t: For wat kernels π(..) does termo. cost increase wit time?

17 COMPLEXITY DYNAMICS OF BIOSPHERE E(cost) = X apple X ( t+1 t )P ( t )ln ( t+1 t ) t, t+1 were t is te obserable s alue at time t; π(..) is te conditional distribution of dynamics; 0 t Plug in π(..) of terrestrial biospere: Does termo. cost of biospere beaior increase wit time?

18 COMPLEXITY DYNAMICS OF BIOSPHERE E(cost) = X apple X ( t+1 t )P ( t )ln ( t+1 t ) t, t+1 were t is te obserable s alue at time t; π(..) is te conditional distribution of dynamics; 0 t Plug in π(..) of terrestrial biospere: How far is termo. cost of biospere from upper bound of solar free energy flux?