Analogy Between Gambling and Measurement-Based Work Extraction Dror Vinkler 1 Haim Permuter 1 Neri Merhav 2 1 Ben Gurion University 2 Technion - Israel Institute of Technology ISIT 2014 1 / 17
Outline Past results and brief physics background The analogy Gambling on continuous random variables Consequences Universal engine Memory Summary and future work 2 / 17
Past Results - Horse Race Gambling Kelly 1956 A race of m horses. 3 / 17
Past Results - Horse Race Gambling Kelly 1956 A race of m horses. X i - winning horse. Y i - side information. 3 / 17
Past Results - Horse Race Gambling Kelly 1956 A race of m horses. X i - winning horse. Y i - side information. X i , Y i are pairwise i.i.d. 3 / 17
Past Results - Horse Race Gambling Kelly 1956 A race of m horses. X i - winning horse. Y i - side information. X i , Y i are pairwise i.i.d. S n - gambler’s capital after n rounds, S 1 = b X | Y ( X 1 | Y 1 ) o X ( X 1 ) S 0 , b X | Y ( X | Y ) - betting strategy. o X ( X ) - odds. 3 / 17
Past Results - Horse Race Gambling Kelly 1956 A race of m horses. X i - winning horse. Y i - side information. X i , Y i are pairwise i.i.d. S n - gambler’s capital after n rounds, S 1 = b X | Y ( X 1 | Y 1 ) o X ( X 1 ) S 0 , b X | Y ( X | Y ) - betting strategy. o X ( X ) - odds. � n S n = b X | Y ( X i | Y i ) o X ( X i ) S 0 , i =1 3 / 17
Past Results - Horse Race Gambling Kelly 1956 The goal: finding the optimal betting strategy b ∗ X | Y , b ∗ X | Y = arg max E [log S n ] . b X | Y log S n - capital growth. 4 / 17
Past Results - Horse Race Gambling Kelly 1956 The goal: finding the optimal betting strategy b ∗ X | Y , b ∗ X | Y = arg max E [log S n ] . b X | Y log S n - capital growth. This optimal betting strategy is b ∗ X | Y = P X | Y . 4 / 17
Past Results - Horse Race Gambling Kelly 1956 The goal: finding the optimal betting strategy b ∗ X | Y , b ∗ X | Y = arg max E [log S n ] . b X | Y log S n - capital growth. This optimal betting strategy is b ∗ X | Y = P X | Y . A bet is fair if o X ( x ) = 1 /P X ( x ) ∀ x . 4 / 17
Past Results - Horse Race Gambling Kelly 1956 The goal: finding the optimal betting strategy b ∗ X | Y , b ∗ X | Y = arg max E [log S n ] . b X | Y log S n - capital growth. This optimal betting strategy is b ∗ X | Y = P X | Y . A bet is fair if o X ( x ) = 1 /P X ( x ) ∀ x . For a fair bet, the maximal growth is nI ( X ; Y ) . 4 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 X = 2 X = 1 X = 0 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 For a fixed temperature, X = 2 X = 1 P X ( x ) ∼ e −E ( x ) /k B T . X = 0 P X ( x ) - the Boltzmann distribution. 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 For a fixed temperature, X = 2 X = 1 P X ( x ) ∼ e −E ( x ) /k B T . X = 0 P X ( x ) - the Boltzmann distribution. 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 For a fixed temperature, X = 2 X = 1 P X ( x ) ∼ e −E ( x ) /k B T . X = 0 P X ( x ) - the Boltzmann distribution. Principle 2: 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 For a fixed temperature, X = 2 X = 1 P X ( x ) ∼ e −E ( x ) /k B T . X = 0 P X ( x ) - the Boltzmann distribution. Principle 2: The second law of thermodynamics: ∆ S ≥ 0 . S - the entropy of the system. 5 / 17
Brief Background on Statistical Mechanics For large systems, classical analysis is not possible. Principle 1: X - state of particle. E X = 4 X = 3 For a fixed temperature, X = 2 X = 1 P X ( x ) ∼ e −E ( x ) /k B T . X = 0 P X ( x ) - the Boltzmann distribution. Principle 2: The second law of thermodynamics: ∆ S ≥ 0 . S - the entropy of the system. In a complete cycle: E [ W ] ≤ 0 . 5 / 17
Past Results - Measurement-Based Work Extraction Originated with Maxwell’s demon in the 19 th century. 6 / 17
Past Results - Measurement-Based Work Extraction Originated with Maxwell’s demon in the 19 th century. Using fluctuation theorems, it was shown that [Sagawa and Ueda 2010] E [ W ] ≤ k B TI ( X ; Y ) . 6 / 17
Past Results - Measurement-Based Work Extraction Originated with Maxwell’s demon in the 19 th century. Using fluctuation theorems, it was shown that [Sagawa and Ueda 2010] E [ W ] ≤ k B TI ( X ; Y ) . Some systems achieve this, but no general achievability scheme. 6 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V = 1 b d c 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V 0 ( X ) V = 1 d c 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V 0 ( X ) V = 1 d Y 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V 0 ( X ) V f ( X ) V = 1 Y 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V f ( X ) V 0 ( X ) V = 1 Y 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V f ( X ) V 0 ( X ) V = 1 Y X - the particle’s location. Y - a noisy measurement. 7 / 17
Past Results - The Szilard Engine Presented by Sagawa and Ueda as an example: V f ( X ) V 0 ( X ) V = 1 Y X - the particle’s location. Y - a noisy measurement. Extracted work is W = k B T ln V f ( X | Y ) V 0 ( X ) . 7 / 17
Past Results - The Szilard Engine Optimal value of V f ( X | Y ) is V ∗ f ( X | Y ) = P X | Y ( X | Y ) . 8 / 17
Past Results - The Szilard Engine Optimal value of V f ( X | Y ) is V ∗ f ( X | Y ) = P X | Y ( X | Y ) . Maximal extracted work is max E [ W ] = k B TI ( X ; Y ) . 8 / 17
Past Results - The Szilard Engine Optimal value of V f ( X | Y ) is V ∗ f ( X | Y ) = P X | Y ( X | Y ) . Maximal extracted work is max E [ W ] = k B TI ( X ; Y ) . Upper bound is achieved. 8 / 17
Past Results - The Szilard Engine Optimal value of V f ( X | Y ) is V ∗ f ( X | Y ) = P X | Y ( X | Y ) . Maximal extracted work is max E [ W ] = k B TI ( X ; Y ) . Upper bound is achieved. Result is specific to one particle of ideal gas. 8 / 17
Main Results Gambling Maxwell’s Demon Side information Measurements results 9 / 17
Main Results Gambling Maxwell’s Demon Side information Measurements results o X - odds 1 /V 0 - initial vol. 9 / 17
Main Results Gambling Maxwell’s Demon Side information Measurements results o X - odds 1 /V 0 - initial vol. b X | Y - betting strategy V f - final vol. 9 / 17
Main Results Gambling Maxwell’s Demon Side information Measurements results o X - odds 1 /V 0 - initial vol. b X | Y - betting strategy V f - final vol. log S n - log of capital W n - extracted work 9 / 17
Main Results Gambling Maxwell’s Demon Side information Measurements results o X - odds 1 /V 0 - initial vol. b X | Y - betting strategy V f - final vol. log S n - log of capital W n - extracted work � n � n k B T ln V f ( X i | Y i ) log S n = log b X | Y ( X i | Y i ) o X ( X i ) , W n = V 0 ( X i ) . i =1 i =1 9 / 17
Some Immediate Consequences o X ⇔ 1 /V 0 = 1 /P X 10/ 17
Some Immediate Consequences o X ⇔ 1 /V 0 = 1 /P X ⇒ In physics, the bet is always fair. 10/ 17
Some Immediate Consequences o X ⇔ 1 /V 0 = 1 /P X ⇒ In physics, the bet is always fair. Second law of thermo. - without measurements E [ W ] ≤ 0 . 10/ 17
Some Immediate Consequences o X ⇔ 1 /V 0 = 1 /P X ⇒ In physics, the bet is always fair. Second law of thermo. - without measurements E [ W ] ≤ 0 . Enables straightforward generalization to m dividers V ∗ f ( X | Y ) = P X | Y ( X | Y ) . 10/ 17
Recap What we achieved so far: 11/ 17
Recap What we achieved so far: One-to-one mapping. 11/ 17
Recap What we achieved so far: One-to-one mapping. Specific to Szilard engine. 11/ 17
Continuous Random Variables - Physics Horowitz and Parrondo 2011, Esposito and Van den Broeck 2011 A single particle in a potential field: E 0 f X x 0 12/ 17
Continuous Random Variables - Physics Horowitz and Parrondo 2011, Esposito and Van den Broeck 2011 A single particle in a potential field: f X f X | y E 0 E 0 Y x x 0 0 12/ 17
Continuous Random Variables - Physics Horowitz and Parrondo 2011, Esposito and Van den Broeck 2011 A single particle in a potential field: E f f X f X | y f X | y E 0 E 0 E 0 → E f Y x x x 0 0 0 12/ 17
Continuous Random Variables - Physics Horowitz and Parrondo 2011, Esposito and Van den Broeck 2011 A single particle in a potential field: E f f X f X | y f X | y E 0 E 0 E 0 → E f Y x x x 0 0 0 12/ 17
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