Thermodynamics: Systems, Properties, and Equilibrium

Thermodynamics Introduction

• Thermodynamics examines energy, its interconversions, and relationships between heat, work and energy.
• Energy transfer occurs via heat (due to temperature differences) or work (not due to temperature differences).
• A system is a specific portion of matter under study, separated from the surroundings by a boundary.
• The boundary can be fixed or movable.

Types of Thermodynamic Systems

• Closed System: Exchanges energy but not matter.
• Open System: Exchanges both energy and matter.
• Isolated System: Exchanges neither energy nor matter.

Thermodynamic Properties

• Intensive properties: independent of system's mass (e.g., temperature, pressure, density).
• Extensive properties: dependent on system's mass (e.g., mass, volume, energy).
• System's state is defined by the values of its properties; any change in property values changes the state.
• Thermodynamic equilibrium needs thermal (uniform temperature), mechanical (no pressure change), and chemical (no compositional change) equilibrium.
• A process is any change a system undergoes; the series of states it passes through is the path.

Types of Thermodynamic Processes

• Isothermal: constant temperature.
• Isobaric: constant pressure.
• Isochoric: constant volume.
• Adiabatic: no heat transfer.
• Cyclic: returns to initial state.

Zeroth Law of Thermodynamics

• If two bodies are each in thermal equilibrium with a third body, they are in thermal equilibrium with each other.

Temperature Scales

• Celsius: water freezes at 0°C, boils at 100°C.
• Fahrenheit: water freezes at 32°F, boils at 212°F.
• Kelvin: absolute zero at 0 K; K = °C + 273.15.
• Rankine: absolute zero at 0 R; R = °F + 459.67.

Temperature Scale Conversion Formulas

• $T(^\circ F) = T(^\circ C) \cdot \frac{9}{5} + 32$
• $T(^\circ C) = (T(^\circ F) - 32) \cdot \frac{5}{9}$
• $T(R) = T(^\circ F) + 459.67$
• $T(K) = T(^\circ C) + 273.15$
• $T(K) = T(R) \cdot \frac{5}{9}$
• $T(R) = T(K) \cdot \frac{9}{5}$

First Law of Thermodynamics

• Energy is conserved; it can only change forms, based on the conservation of energy. The total energy of a system can change due to energy entering or leaving.
• The change in the system's total energy ($\Delta E_{system}$) equals the energy entering ($E_{in}$) minus the energy leaving ($E_{out}$): $\Delta E_{system} = E_{in} - E_{out}$.

Forms of Energy

• Kinetic Energy (KE): Energy of motion ($KE = \frac{1}{2}mv^2$).
• Potential Energy (PE): Energy of position ($PE = mgh$).
• Internal Energy (U): Sum of microscopic energies within a system ($\Delta U = m c_v \Delta T$).

Energy Transfer: Heat (Q) and Work (W)

• Heat (Q): Energy transfer due to temperature difference ($Q = mc\Delta T$). $c$ is specific heat.
• Work (W): Energy transfer not due to temperature difference ($W = \int_{1}^{2} P dV$).
• For a constant pressure process: $W = P(V_2 - V_1)$

First Law for Closed Systems

• $\Delta U + \Delta KE + \Delta PE = Q - W$
• For stationary systems (ΔKE = ΔPE = 0): $\Delta U = Q - W$
• In a cyclic process (initial and final states are the same, so ΔU = 0): $Q = W$ or $\oint \delta Q = \oint \delta W$

Enthalpy (H)

• Defined as $H = U + PV$; change in enthalpy is $\Delta H = \Delta U + \Delta (PV)$.
• At constant pressure: $\Delta H = \Delta U + P\Delta V$ and $\Delta H = Q_p$ where $Q_p$ is the heat transferred at constant pressure.
• Specific heat at constant pressure: $c_p = \left( \frac{\partial h}{\partial T} \right)_p$
• Specific heat at constant volume: $c_v = \left( \frac{\partial u}{\partial T}right)_v$
• Relationship between $c_p$ and $c_v$ for ideal gases: $c_p = c_v + R$, where $R$ is the gas constant.

First Law for Open Systems

• Mass balance: $\Delta m_{system} = m_{in} - m_{out}$, or as a rate: $\frac{dm}{dt} = \sum \dot{m_i} - \sum \dot{m_e}$

Conservation of Energy

• $\frac{dE}{dt} = \dot{Q} - \dot{W} + \sum \dot{m_i} h_i - \sum \dot{m_e} h_e$
• Steady-Flow Process: Fluid moves steadily through a volume, properties change with position, not time.
• For steady flow, mass flow rate in equals mass flow rate out ($\sum \dot{m_i} = \sum \dot{m_e}$).

Steady-Flow Process Conservation of Energy

• For steady flow, $\frac{dE}{dt} = 0$ implies $\dot{Q} - \dot{W} + \sum \dot{m_i} h_i = \sum \dot{m_e} h_e$

Common Steady-Flow Device Analysis

• Nozzles and Diffusers (accelerate/decelerate fluid): $\dot{Q} = 0$, $\dot{W} = 0$, $\Delta PE = 0$ so $h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$
• Turbines (extract energy to produce power): $\dot{Q} = 0$, $\Delta KE = 0$, $\Delta PE = 0$ so $\dot{W} = \dot{m}(h_1 - h_2)$
• Compressors (increase fluid pressure): $\dot{Q} = 0$, $\Delta KE = 0$, $\Delta PE = 0$ so $\dot{W} = \dot{m}(h_1 - h_2)$
• Throttling Valves (cause pressure drop): $\dot{Q} = 0$, $\dot{W} = 0$, $\Delta KE = 0$, $\Delta PE = 0$ so $h_1 = h_2$
• Heat Exchangers (transfer heat between fluids): $\dot{W} = 0$, $\Delta KE = 0$, $\Delta PE = 0$ so $\dot{Q} = \dot{m}(h_e - h_i)$

Second Law of Thermodynamics Overview

• Processes proceed in a specific direction, satisfying both the first and second laws; processes won't happen unless both laws are satisfied.

Thermal Energy Reservoirs

• Hypothetical bodies with large thermal capacity; can supply/absorb finite heat without changing temperature.

Heat Engines

• Devices converting heat into work; receive heat from a high-temperature source, convert part to work, reject remaining waste heat. Efficiency is the net work output to the heat input ratio, $\eta_{th} = \frac{W_{net, out}}{Q_{in}}$ = 1 - $\frac{Q_{out}}{Q_{in}}$
• Refrigerators and Heat Pumps: Devices transferring heat from a low-temperature source to a high-temperature sink.

Coefficient of Performance (COP)

• Refrigerator: $COP_R = \frac{Q_L}{W_{net, in}}$
• Heat Pump: $COP_{HP} = \frac{Q_H}{W_{net, in}}$
• Relationship: $COP_{HP} = COP_R + 1$

Thermodynamic Processes

• Reversible processes leave no trace on surroundings; system and surroundings return to initial states.
• Irreversible processes cannot reverse without leaving a trace.
• Irreversibilities originate from friction, non-equilibrium expansion/compression, heat transfer through ΔT, mixing, electrical resistance, inelastic deformation, and chemical reactions.

Carnot Cycle

• An efficient theoretical cycle for converting heat into work, or work into heat (refrigeration).
• Carnot Cycle Steps: Isothermal Expansion, Adiabatic Expansion, Isothermal Compression, Adiabatic Compression

Carnot Principles

• Irreversible heat engine efficiency is always less than that of a reversible one between the same reservoirs.
• Efficiencies are equal for all reversible heat engines operating between same reservoirs.

Carnot Efficiency Equations

• Carnot heat engine: $\eta_{th, Carnot} = 1 - \frac{T_L}{T_H}$
• Carnot refrigerator: $COP_{R, Carnot} = \frac{1}{\frac{T_H}{T_L} - 1}$
• Carnot heat pump: $COP_{HP, Carnot} = \frac{1}{1 - \frac{T_L}{T_H}}$

Entropy Basics

• Entropy is a measure of a system's disorder or randomness; higher entropy equals greater disorder.

Entropy Change and Generation

• Reversible process change: $dS = \frac{\delta Q}{T}$
• Irreversible process change: $dS > \frac{\delta Q}{T}$
• Entropy generation ($S_{gen}$) measures irreversibility; always positive or zero; zero only for reversible processes: $S_{gen} = \Delta S_{system} - \int \frac{\delta Q}{T} \geq 0$

Entropy Increase Principle

• The total entropy of an isolated system always increases or remains constant in reversible process.
• $\Delta S_{isolated} \geq 0$

Ideal Gas Entropy

• Constant Specific Heats:
  • $\Delta s = c_v \ln{\frac{T_2}{T_1}} + R \ln{\frac{v_2}{v_1}}$
  • $\Delta s = c_p \ln{\frac{T_2}{T_1}} - R \ln{\frac{P_2}{P_1}}$
• Variable Specific Heats: $\Delta s = s_2^\circ - s_1^\circ - R \ln{\frac{P_2}{P_1}}$

Isentropic Process

• Entropy remains constant: $s_1 = s_2$

Ideal Gas Isentropic Relations

• Constant Specific Heats:
  • $(\frac{T_2}{T_1}) = (\frac{v_1}{v_2})^{k-1}$
  • $(\frac{T_2}{T_1}) = (\frac{P_2}{P_1})^{\frac{k-1}{k}}$
  • $(\frac{P_2}{P_1}) = (\frac{v_1}{v_2})^{k}$
• $k = \frac{c_p}{c_v}$ is the specific heat ratio.
• Variable Specific Heats:
  • $(\frac{P_2}{P_1}) = \frac{P_{r2}}{P_{r1}}$
  • $(\frac{v_2}{v_1}) = (\frac{v_{r2}}{v_{r1}})$

ALGORITHMIC GAME THEORY - LECTURE 7: THE PRICE OF ANARCHY

Social Welfare

• Social Welfare equation: $$SW(a) = \sum_{i \in N} u_i(a)$$.
• Optimal Social Welfare equation: $$SW(a^*) = \max_{a \in A} SW(a)$$.

Price of Anarchy (PoA) and Stability (PoS)

• Price of Anarchy (PoA) equation: $$PoA = \frac{\max_{a \in A} SW(a)}{\min_{a \in S} SW(a)} = \frac{SW(a^*)}{SW(a^{worst})}$$.
• Price of Stability (PoS) equation: $$PoS = \frac{\max_{a \in A} SW(a)}{\max_{a \in S} SW(a)} = \frac{SW(a^*)}{SW(a^{best})}$$.
• $PoA > PoS > 1$

Braess Paradox Setting

• A directed graph $G = (V, E)$.
• Two players want to travel from $s$ to $t$, each player controls one unit of traffic.
• The latency (or cost) on each edge depends on the amount of traffic.

Braess Paradox Example

• Without the edge (v, w): paths $s \rightarrow v \rightarrow t$ and $s \rightarrow w \rightarrow t$, Social Cost: $4$
• With the edge (v, w): path is: $s \rightarrow v \rightarrow w \rightarrow t$

Load Balancing Game Setting

• Assign $n$ jobs to $m$ machines, where job $i$ has weight $w_i$ and machine $j$ has speed $s_j$.
• Cost for job $i$ equation: job $i$ is$$c_i(a) = \frac{\sum_{j:a_j = a_i} w_j}{s_{a_i}}$$

Load Balancing Social Cost

• Makespan is maximum load on any machine, Social Cost equation: $$SC(a) = \max_{j} \frac{\sum_{i:a_i = j} w_i}{s_j}$$.

Research Methods: Core Concepts

Populations and Samples Definition

• Population: The entire group of interest in a study.
• Sample: A subset of the population selected for study that is representative sample, is accurate reflection of population.
• Sampling Frame: A list of population members from which the sample is drawn.
• Sampling Unit: An individual member of the sampling frame.

Sample Size Equation

• Probability sample size is defined as: $$$$\text{Probability sample size} = \frac{\text{k (measure of variability)} \cdot (\text{z-score corresponding to desired confidence})^2}{(\text{acceptable margin of error})^2}$$$$
• where $k = 1$ for samples and $k > 1$ when comparing 2 or more subgroups.

Sampling Methods Types

• Simple Random Sample: Each member has an equal chance of selection.
• Systematic Sample: Selection at regular intervals.
• Stratified Sample: Random samples from subgroups (strata).
• Cluster Sample: Random sample of clusters.
• Convenience Sample: Readily available sample.
• Quota Sample: Sample that reflects the characteristics of the population.
• Snowball Sample: Participants recruit other participants.

Variable Types

• Categorical: Values fall into distinct categories. Nominal: Unordered (gender, eye color). Ordinal: Ordered (education level, satisfaction).
• Numerical: Values represent quantities. Discrete: Whole numbers (children, cars). Continuous: Any value within a range (height, weight).
• Independent Variable: Manipulated or changed by the researcher.
• Dependent Variable: Measured in response to the independent variable.

Hypothesis Testing Overview

• Null Hypothesis: A statement of no effect or no difference.
• Alternative Hypothesis: Contradicts the null hypothesis.

Errors in Hypothesis Testing

• Type I Error (False Positive): Rejecting a true null hypothesis.
• Type II Error (False Negative): Failing to reject a false null hypothesis.

P-Value and testing

• P-Value: Probability of results as extreme as observed, assuming the null hypothesis is true.
• Significance Level (Alpha): Threshold for rejecting the null hypothesis (typically 0.05).
• If the p-value is less than or equal to the alpha is rejected.

Using Tests

• T-Test: Compares means of two groups.
• ANOVA: Compares means of three or more groups.

Correlation vs. Causation

• Correlation: Statistical association between variables.
• Causation: One variable directly causes a change in another.
• Note: Correlation does not imply causation.

Research Study Designs

• Descriptive Studies: Describe population characteristics.
• Correlational Studies: Examine relationships between variables.
• Experimental Studies: Manipulate variables.
  • Control Group: Does not receive treatment.
  • Random Assignment: Participants are randomly assigned.

Bias Control

• Blind Study: Participants unaware of treatment status.
• Double-Blind Study: Both participants and researchers unaware.

Validity vs Reliability

• Reliability: Consistency or repeatability of a measure.
• Validity: Accuracy; measure reflects intended concept.
• Note: Reliable measures aren't valid, and to be valid must be Reliable

Lecture 10 Highlights

• A lecture about two ways to sort items: Quick Sort and Linear Time Sorting

Quick Sort

• Sorting technique that uses divide and conquer to divide an array to sort items. Then combines sub arrays to form a final result

Divide aspect

• Array is rearranges into 2 sub-arrays
• A pivot $x$is used
• Elements in the left subarray $\le x$
• Elements in the right subarray $\ge x$

Conquer aspect

• Recursively sorts each sub-array : 2 $T(n/2)$

Combine aspect

• In place
• $Θ(n lg n)$ "on average"

Important pivot consideration

Choice of pivot is important!

1. First element

• O(n2) if array is already sorted

2. Random element

• O(n lg n) "on average"

3. Median

• O(n lg n) guaranteed, but hard to compute

4. "Median of three"

• often works well in practice
• take median of: First, last, middle

Performance

• Worst Case: $Θ(n2)$
• e.g., input already sorted (and we pick first element as pivot)
• Best Case: $Θ(n lg n)$
• e.g., pivot always in middle
• Average Case: $Θ(n lg n)$

Randomization

• Idea: Partition around a random element
• Ensures no particular input elicits worst-case behavior
• Random choice within PARTITION procedure
• Expected running time: $O(n lg n)$

Linear Time Sorting

• Counting sort, Radix sort, Bucket sort
• Comparison Sorts:
• Merge Sort, Quick Sort, Heap Sort...
• Make decisions based on $\le, \ge, = $
• Any comparison sort requires $Ω(n lg n)$ comparisons in the worst case

Counting Sort

• Assumptions:
• n input elements
• Each element is an integer between 0 and k, for some integer k. $k = O(n)$

Basics

• Determine, for each input element x, the number of elements < x.
• Place x directly into its position.

Absolute Value Function Basics

• Absolute Value Function description: $$f(x)= |x| = \begin{cases} x, , \text{si } x \geq 0 \
• x, , \text{si } x < 0 \end{cases}$$
• Absolute Value Function examples:

- |5| = 5
- |-5| = -(-5) = 5
- |0| = 0

What an absolute value looks like graphed

• The function is always positive or zero.
• The function is symmetric about the y axis.
• The function is decreasing for $x < 0$ and increasing for $x > 0$.

Value transformations

$$f(x) = a|b(x - h)| + k$$

• Where:
• $a$ is the vertical stretch
• $b$ is the horizontal stretch
• $h$ is the horizontal translation
• $k$ is the vertical translation

Value transformation Example

$$f(x) = 2|x - 3| + 1$$.

• The vertex of the function is the point (3, 1).

Solving Value equations

Equations with the absolute value must consider equation positive or negative.

1. Positive case = 0
2. Negative case = 0

Solving Value equations example-

• Let's solve:* $|x - 2| = 3$.
• Cas 1: $x - 2 \geq 0$. Solution = 5.
• Cas 2: $x - 2 < 0$. Solution = -1.

Solving Value example is $x = 5$ and $x = -1$.

In summary, the absolute value function is a function that gives the distance of a number to zero. It is always positive or zero and can be transformed

Wave Equations Recap

Disturbance Analysis

• Small disturbances $\eta(x,t)$ to a stretched string obey the equation equation: $\frac{\partial^2 \eta}{\partial t^2} = v^2 \frac{\partial^2 \eta}{\partial x^2}$ The wave is $v = \sqrt{\frac{T}{\rho}}$ is the wave speed, $T$ is the tension in the string, $\rho$ is the mass per unit length.

Linearity of function

Equation is linear : if $\eta_1(x,t)$ and $\eta_2(x,t)$ are solutions, then so is $A\eta_1(x,t) + B\eta_2(x,t)$ with $A,B$ constants.

General solution

General solution is equation: $\eta(x,t) = f(x-vt) + g(x+vt)$

Harmonic waves

Hormonic waves example equation is: $\eta(x,t) = A \cos(kx - \omega t)$

• Where:
• A is the amplitude of the wave
• k is the wave number, $k = \frac{2\pi}{\lambda}$, with λ the wavelength
• ω is the angular frequency, ω = 2πf, with f the frequency (number of oscillations per unit time).

Complex representation

Complex wave equation: $\eta(x,t) = A e^{i(kx - \omega t)}$

Superposition

Equations

• Example waves: - $\eta_1(x,t) = A_1 \cos(kx - \omega t)$ and $\eta_2(x,t) = A_2 \cos(kx - \omega t + \phi)$ for phase difference Φ
• Rewrite as - $\eta(x,t) = A \cos(kx - \omega t + \theta)$ The equations are -
• A = √(($A_1$ + $A_2$ cosΦ)^2 + ($A_2$sinΦ)^2)
• tanΘ =($A_2$sinΦ)/($A_1$ + $A_2$ CosΦ)

Special cases

• If Φ = 0 , the waves have the equation A = $A_1$ + $A_2$
• If Φ = π the waves have the equation A = |$A_1 + A_2$|

Static Games of Complete Information

Game Definition

• What to contain at minimum: > 2 players, each player set of options, payoff depends on total choice

Static Game Definition

• Players move at the same time where each players moves are possible with their own payoff known .

Game Representation

• Normal Form representation: all players, each with a strategy space and payoff function (equation: $G = {S_1,...,S_n;u_1,...,u_n}$).
• Extensive Form (Game Tree)

Prisoner's Dilemma Case Study

• Players: Two suspects.

• Strategies: Confess or Not Confess.

• Payoffs: (Years in prison)

  |          | Player 2 Confesses | Player 2 Doesn't Confess |
  | :------- | :----------------: | :-----------------------: |
  | Player 1 Confesses    |       -5, -5        |           0, -10      |
  | Player 1 Doesn't Confess |       -10, 0       |           -1, -1        |
  

Dominant Strategy Example

• (Player 1 confeses, Confess,Player 2 Doesn't Confess).

Assumptions

• Players are rational: they want to maximize their own payoff
• Participants are intelligent: they know everything about the game

Strictly Dominant Strategy Definition

• Strategy $s'i$ is strictly dominant if equation: $u_i(s'i,s{-i}) > u_i(s''i,s{-i}), \forall s{-i} \in S_{-i}$

Iterated Elimination - Strictly Dominated Strategies definition

• For each player i, eliminate all strictly dominated strategies from Sk-1 to get Sk.

• Outcome: (Middle, Center)

Quantum Mechanics

Prerequisite Knowledge

• Commitment to learning of the principles.
• Comfort with calculus basics, vectors and matrices, complex numbers.
• Classical mechanics as a bonus.

Basic Principles for you learning

• Basic Principles
• The calculation of anything
• 10 lectures

What is in the 10 lectures

• Lecture 1: Beginnings:
• Rule of probability,
• Action of distance,
• experiment,
• Two-Slit,
• amplitudes.
• Lecture 2: Vector Spaces:
• Bra-ket notation,
• Eigen and Functions.
• Lecture 3: Operators:
• Functions of operators,
• Commutators,
• Position and momentum.
• Lecture 4: Dynamics:
• Time-translation operator,
• Harmonic oscillator,
• Schrodinger,
• Superposition.
• Lecture 5: Quantal Mechanics:
• density operator.
• Mechanics
• chaos ,
• ensembles and evolution.
• Lecture 6: Symmetry:
• Time reversal,
• Rotations,
• Translations.
• Lecture 7: Identical Particles:
• Quantum field theory,
• Helium atom,
• Two particle stages
• Lecture 8: Perturbation Theory:
• Fermi's Golden Rule,
• Hamiltonians ,
• Hydrogen ionization.
• Lecture 9: quantum Issues
• The Many interpretation,
• Decoherence,
• Measurements
• Lecture 10: After