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Kernel: Julia 1.11

Modern Thermodynamics: From Stars to Biology

Learning Objectives

By the end of this tutorial, you will:

Apply thermodynamics to stellar astrophysics and black hole physics
Understand biological thermodynamics and metabolic processes
Analyze modern energy systems and climate applications
Explore cutting-edge applications in nanotechnology and quantum systems
Connect thermodynamic principles to information theory
Utilize Julia for advanced thermodynamic modeling
Master CoCalc collaboration for interdisciplinary research

Expanding Thermodynamic Horizons

This notebook explores how thermodynamic principles extend far beyond traditional engineering applications. From the nuclear furnaces of stars to the molecular machinery of living cells, thermodynamics provides a universal framework for understanding energy and entropy across all scales.

Historical Context: The 20th century saw thermodynamics expand into new realms—statistical mechanics explained molecular behavior, astrophysics revealed stellar thermodynamics, and biochemistry uncovered the energy basis of life.

CoCalc Advantage: Interdisciplinary collaboration between physicists, biologists, engineers, and climate scientists, enabling breakthrough research at the intersections of traditional fields.

In [9]:

import Pkg; Pkg.add("PlotThemes")

using Plots, Printf, Statistics, LinearAlgebra
using PlotThemes
theme(:bright)

# Fundamental constants (2018 CODATA values)
const k_B = 1.380649e-23      # Boltzmann constant (J/K)
const N_A = 6.02214076e23     # Avogadro's number (1/mol)
const R = k_B * N_A           # Universal gas constant (J/mol·K)
const σ_SB = 5.670374419e-8   # Stefan-Boltzmann constant (W/m²·K⁴)
const c = 299792458.0         # Speed of light (m/s)
const h = 6.62607015e-34      # Planck constant (J·s)
const G = 6.67430e-11         # Gravitational constant (m³/kg⋅s²)
const M_sun = 1.989e30        # Solar mass (kg)

println("🌟 MODERN THERMODYNAMICS: FROM STARS TO BIOLOGY")
println(repeat("=", 55))
println("Exploring thermodynamics across scales and disciplines")
println()
println("Universal Constants:")
println("• Boltzmann constant: k_B = $(k_B) J/K")
println("• Stefan-Boltzmann: σ = $(σ_SB) W/(m²·K⁴)")
println("• Gravitational constant: G = $(G) m³/(kg⋅s²)")
println("• Solar mass: M☉ = $(M_sun) kg")

Out[9]:

   Resolving package versions...
  No Changes to `~/.julia/environment/v1.11/Project.toml`
  No Changes to `~/.julia/environment/v1.11/Manifest.toml`

🌟 MODERN THERMODYNAMICS: FROM STARS TO BIOLOGY
=======================================================
Exploring thermodynamics across scales and disciplines

Universal Constants:
• Boltzmann constant: k_B = 1.380649e-23 J/K
• Stefan-Boltzmann: σ = 5.670374419e-8 W/(m²·K⁴)
• Gravitational constant: G = 6.6743e-11 m³/(kg⋅s²)
• Solar mass: M☉ = 1.989e30 kg

1. Stellar Thermodynamics and Astrophysics

Stars as Thermodynamic Systems

Stars are massive thermodynamic systems governed by:

Hydrostatic equilibrium: Pressure gradient balances gravitational force
Energy transport: Radiation and convection carry energy outward
Nuclear fusion: Converts mass to energy in stellar cores
Stellar evolution: Thermodynamic properties change over stellar lifetime

Key Stellar Relations

Stefan-Boltzmann Law for stellar luminosity: $L = 4\pi R^2 \sigma T_{eff}^4$

where $L$ is luminosity, $R$ is stellar radius, and $T_{eff}$ is effective temperature.

Hydrostatic Equilibrium: $\frac{dP}{dr} = -\frac{\rho GM(r)}{r^2}$

Virial Theorem for gravitational systems: $2K + U = 0$

where $K$ is kinetic energy and $U$ is gravitational potential energy.

Black Hole Thermodynamics

Hawking Temperature: $T_H = \frac{\hbar c^3}{8\pi G M k_B}$

Bekenstein-Hawking Entropy: $S_{BH} = \frac{A c^3}{4G\hbar}$

where $A = 4\pi r_s^2$ is the event horizon area and $r_s = 2GM/c^2$ is the Schwarzschild radius.

In [10]:

# Stellar physics functions
function stellar_luminosity(R, T_eff)
    """Stefan-Boltzmann law: L = 4πR²σT⁴"""
    return 4π * R^2 * σ_SB * T_eff^4
end

function main_sequence_lifetime(M)
    """Rough main sequence lifetime: τ ∝ M/L ∝ M^(-2.5) (very approximate)"""
    # For solar-type stars, lifetime scales as (M/M_sun)^(-2.5)
    τ_sun = 10e9 * 365.25 * 24 * 3600  # 10 billion years in seconds
    return τ_sun * (M / M_sun)^(-2.5)
end

function hawking_temperature(M)
    """Hawking temperature: T_H = ℏc³/(8πGMk_B)"""
    ℏ = h / (2π)
    return (ℏ * c^3) / (8π * G * M * k_B)
end

function schwarzschild_radius(M)
    """Schwarzschild radius: r_s = 2GM/c²"""
    return 2 * G * M / c^2
end

function bekenstein_hawking_entropy(M)
    """Bekenstein-Hawking entropy: S = Ac³/(4Gℏ)"""
    ℏ = h / (2π)
    r_s = schwarzschild_radius(M)
    A = 4π * r_s^2  # Event horizon area
    return A * c^3 / (4 * G * ℏ)
end

println("=== STELLAR THERMODYNAMICS ===")
println("Main Sequence Stars (Hydrostatic Equilibrium):")
println()

# Main sequence stars data
stellar_data = [
    ("Red dwarf (M-type)", 0.1, 0.1, 3000),
    ("Sun (G-type)", 1.0, 1.0, 5778),
    ("Blue giant (B-type)", 10.0, 5.0, 20000),
    ("Blue supergiant (O-type)", 50.0, 20.0, 40000)
]

println("Main Sequence Properties:")
println(repeat("-", 80))
@printf "%-25s %8s %8s %10s %12s %12s\n" "Star Type" "M/M☉" "R/R☉" "T_eff (K)" "L/L☉" "Lifetime (yr)"
println(repeat("-", 80))

for (star_name, M_ratio, R_ratio, T_eff) in stellar_data
    M_star = M_ratio * M_sun
    R_star = R_ratio * 6.96e8  # Solar radius in meters
    
    L_star = stellar_luminosity(R_star, T_eff)
    L_sun = stellar_luminosity(6.96e8, 5778)  # Solar luminosity
    L_ratio = L_star / L_sun
    
    lifetime = main_sequence_lifetime(M_star)
    lifetime_years = lifetime / (365.25 * 24 * 3600)
    
    @printf "%-25s %8.1f %8.1f %10.0f %12.1f %12.2e\n" star_name M_ratio R_ratio T_eff L_ratio lifetime_years
end

println()
println("Stellar Evolution Insights:")
println("• More massive stars are hotter, brighter, but shorter-lived")
println("• Luminosity scales roughly as M³⁻⁴ (mass-luminosity relation)")
println("• Lifetime scales as M/L ∝ M⁻²·⁵ (more massive = faster burning)")
println("• Red dwarfs will outlive the universe's current age")
println("• Massive stars explode as supernovae after millions of years")
println()

Out[10]:

=== STELLAR THERMODYNAMICS ===
Main Sequence Stars (Hydrostatic Equilibrium):

Main Sequence Properties:
--------------------------------------------------------------------------------
Star Type                     M/M☉     R/R☉  T_eff (K)         L/L☉ Lifetime (yr)
--------------------------------------------------------------------------------
Red dwarf (M-type)             0.1      0.1       3000          0.0     3.16e+12
Sun (G-type)                   1.0      1.0       5778          1.0     1.00e+10
Blue giant (B-type)           10.0      5.0      20000       3588.8     3.16e+07
Blue supergiant (O-type)      50.0     20.0      40000     918734.0     5.66e+05

Stellar Evolution Insights:
• More massive stars are hotter, brighter, but shorter-lived
• Luminosity scales roughly as M³⁻⁴ (mass-luminosity relation)
• Lifetime scales as M/L ∝ M⁻²·⁵ (more massive = faster burning)
• Red dwarfs will outlive the universe's current age
• Massive stars explode as supernovae after millions of years

In [11]:

using Printf

println("=== BLACK HOLE THERMODYNAMICS ===")
println("Hawking Radiation and Event Horizon Physics")
println()

black_holes = [
    ("Stellar-mass BH", 10.0),
    ("Intermediate BH", 1000.0),
    ("Sagittarius A*", 4e6),
    ("M87* (imaged BH)", 6.5e9),
    ("Galaxy cluster BH", 1e10)
]

println("Black Hole Thermodynamic Properties:")
println(repeat("-", 90))
@printf "%-20s %12s %12s %15s %15s\n" "Black Hole Type" "Mass (M☉)" "r_s (km)" "T_Hawking (K)" "Entropy (J/K)"
println(repeat("-", 90))

for (bh_name, M_bh_solar) in black_holes
    M_bh = M_bh_solar * M_sun
    r_s = schwarzschild_radius(M_bh)
    T_hawk = hawking_temperature(M_bh)
    S_bh = bekenstein_hawking_entropy(M_bh)
    @printf "%-20s %12.1e %12.1f %15.2e %15.2e\n" bh_name M_bh_solar (r_s/1000) T_hawk S_bh
end

println()
println("Black Hole Thermodynamic Insights:")
println("• Larger black holes are colder (T ∝ 1/M)")
println("• Entropy scales with area, not volume (S ∝ M²)")
println("• Stellar-mass BHs evaporate faster than universe age")
println("• Supermassive BHs will outlive all stars")
println("• Information paradox: where does information go?")
println("• Holographic principle: 3D information stored on 2D surface")
println()

println("Black Hole Evaporation:")
M_stellar = 10 * M_sun
τ_evap_years = 2e67 * (M_stellar / M_sun)^3
universe_age = 1.38e10

@printf "• Stellar-mass BH (10 M☉) evaporation time: ~%.1e years\n" τ_evap_years
@printf "• Universe current age: %.1e years\n" universe_age
@printf "• Ratio: %.1e times longer than universe age\n" (τ_evap_years / universe_age)
println("• Black holes are effectively eternal on cosmic timescales!")
println()

Out[11]:

=== BLACK HOLE THERMODYNAMICS ===
Hawking Radiation and Event Horizon Physics

Black Hole Thermodynamic Properties:
------------------------------------------------------------------------------------------
Black Hole Type         Mass (M☉)     r_s (km)   T_Hawking (K)   Entropy (J/K)
------------------------------------------------------------------------------------------
Stellar-mass BH           1.0e+01         29.5        6.17e-09        1.05e+79
Intermediate BH           1.0e+03       2954.1        6.17e-11        1.05e+83
Sagittarius A*            4.0e+06   11816506.2        1.54e-14        1.68e+90
M87* (imaged BH)          6.5e+09 19201822607.9        9.49e-18        4.43e+96
Galaxy cluster BH         1.0e+10 29541265550.6        6.17e-18        1.05e+97

Black Hole Thermodynamic Insights:
• Larger black holes are colder (T ∝ 1/M)
• Entropy scales with area, not volume (S ∝ M²)
• Stellar-mass BHs evaporate faster than universe age
• Supermassive BHs will outlive all stars
• Information paradox: where does information go?
• Holographic principle: 3D information stored on 2D surface

Black Hole Evaporation:
• Stellar-mass BH (10 M☉) evaporation time: ~2.0e+70 years
• Universe current age: 1.4e+10 years
• Ratio: 1.4e+60 times longer than universe age
• Black holes are effectively eternal on cosmic timescales!

2. Biological Thermodynamics and Life

Thermodynamics of Living Systems

Living organisms are non-equilibrium thermodynamic systems that:

Maintain low entropy through continuous energy input
Convert chemical energy to work and heat
Create and maintain complex organized structures
Operate far from thermodynamic equilibrium

Cellular Energy Currency: ATP

ATP hydrolysis provides energy for cellular processes: $\text{ATP} + \text{H}_2\text{O} \rightarrow \text{ADP} + \text{P}_i + \text{energy}$

Standard Gibbs free energy: $\Delta G^° = -30.5$ kJ/mol

Cellular conditions: $\Delta G ≈ -50$ to $-65$ kJ/mol

Photosynthesis: Solar Energy Conversion

Overall reaction: $6\text{CO}_2 + 6\text{H}_2\text{O} + \text{light} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2$

Thermodynamic efficiency: ~1-3% (conversion of solar energy to chemical energy)

Metabolic Heat Generation

Basal metabolic rate for mammals: $\text{BMR} \propto M^{3/4}$

where $M$ is body mass (Kleiber's law).

Protein Folding Thermodynamics

Protein folding is driven by:

Enthalpy: Hydrogen bonds, van der Waals interactions
Entropy: Hydrophobic effect, configurational entropy
Free energy minimization: $\Delta G = \Delta H - T\Delta S$

In [12]:

using Printf

const N_A = 6.02214076e23  # 1/mol, Avogadro's number

# Biological thermodynamics functions
function atp_energy_release()
    """Standard ATP hydrolysis energy in J/mol"""
    return 30.5e3  # J/mol (standard conditions)
end

function cellular_atp_energy()
    """ATP hydrolysis energy under cellular conditions in J/mol"""
    return 55e3  # J/mol (typical cellular conditions)
end

function basal_metabolic_rate(mass_kg)
    """Kleiber's law: BMR = 70 * M^0.75 kcal/day for mammals"""
    bmr_kcal_day = 70 * mass_kg^0.75
    bmr_watts = bmr_kcal_day * 4184 / (24 * 3600)  # Convert to watts
    return bmr_watts
end

function photosynthesis_efficiency()
    """Typical photosynthesis efficiency"""
    return 0.02  # 2% average efficiency
end

println("=== BIOLOGICAL THERMODYNAMICS ===")
println("Energy and Entropy in Living Systems")
println()

# ATP energetics
ΔG_ATP_standard = atp_energy_release()
ΔG_ATP_cellular = cellular_atp_energy()
T_body = 37 + 273.15  # K (body temperature)

println("=== CELLULAR ENERGY CURRENCY (ATP) ===")
@printf "• Standard ATP hydrolysis: ΔG° = %.1f kJ/mol\n" (ΔG_ATP_standard / 1000)
@printf "• Cellular conditions: ΔG ≈ %.1f kJ/mol\n" (ΔG_ATP_cellular / 1000)
@printf "• Body temperature: T = %.1f K = %.1f°C\n" T_body (T_body - 273.15)
println()

# Human ATP turnover
human_mass = 70.0  # kg (average adult)
bmr_human = basal_metabolic_rate(human_mass)
daily_energy = bmr_human * 24 * 3600  # J/day
atp_molecules_per_day = daily_energy / ΔG_ATP_cellular * N_A

println("Human Energy Metabolism:")
@printf "• Adult human mass: %.0f kg\n" human_mass
@printf "• Basal metabolic rate: %.0f W\n" bmr_human
@printf "• Daily energy expenditure: %.1f MJ/day\n" (daily_energy / 1e6)
@printf "• ATP molecules hydrolyzed: %.2e molecules/day\n" atp_molecules_per_day
@printf "• ATP turnover rate: %.2e molecules/second\n" (atp_molecules_per_day / (24 * 3600))
@printf "• Equivalent ATP mass: %.0f kg/day (recycled ~1000 times!)\n" (atp_molecules_per_day * 507 / N_A / 1000)
println()

# Metabolic scaling
println("=== METABOLIC SCALING (KLEIBER'S LAW) ===")
animals = [
    ("Mouse", 0.025),
    ("Rat", 0.3),
    ("Cat", 4.0),
    ("Human", 70.0),
    ("Horse", 500.0),
    ("Elephant", 5000.0)
]

println("Metabolic Rate Scaling (BMR ∝ M^0.75):")
println(repeat("-", 60))
@printf "%-15s %10s %12s %15s\n" "Animal" "Mass (kg)" "BMR (W)" "BMR/Mass (W/kg)"
println(repeat("-", 60))

for (animal, mass) in animals
    bmr = basal_metabolic_rate(mass)
    bmr_per_kg = bmr / mass
    @printf "%-15s %10.3f %12.1f %15.2f\n" animal mass bmr bmr_per_kg
end

println()
println("Metabolic Insights:")
println("• Larger animals have lower metabolic rate per unit mass")
println("• Surface area to volume ratio affects heat loss")
println("• Metabolic efficiency improves with size")
println("• Evolutionary advantage of larger size in cold climates")
println()

Out[12]:

=== BIOLOGICAL THERMODYNAMICS ===
Energy and Entropy in Living Systems

=== CELLULAR ENERGY CURRENCY (ATP) ===
• Standard ATP hydrolysis: ΔG° = 30.5 kJ/mol
• Cellular conditions: ΔG ≈ 55.0 kJ/mol
• Body temperature: T = 310.1 K = 37.0°C

Human Energy Metabolism:
• Adult human mass: 70 kg
• Basal metabolic rate: 82 W
• Daily energy expenditure: 7.1 MJ/day
• ATP molecules hydrolyzed: 7.76e+25 molecules/day
• ATP turnover rate: 8.98e+20 molecules/second
• Equivalent ATP mass: 65 kg/day (recycled ~1000 times!)

=== METABOLIC SCALING (KLEIBER'S LAW) ===
Metabolic Rate Scaling (BMR ∝ M^0.75):
------------------------------------------------------------
Animal           Mass (kg)      BMR (W) BMR/Mass (W/kg)
------------------------------------------------------------
Mouse                0.025          0.2            8.52
Rat                  0.300          1.4            4.58
Cat                  4.000          9.6            2.40
Human               70.000         82.0            1.17
Horse              500.000        358.4            0.72
Elephant          5000.000       2015.6            0.40

Metabolic Insights:
• Larger animals have lower metabolic rate per unit mass
• Surface area to volume ratio affects heat loss
• Metabolic efficiency improves with size
• Evolutionary advantage of larger size in cold climates

In [13]:

using Printf

# Photosynthesis analysis
println("=== PHOTOSYNTHESIS: SOLAR ENERGY CONVERSION ===")

solar_irradiance = 1000  # W/m² (peak sunlight)
photosynthesis_eff = photosynthesis_efficiency()
leaf_areas = [0.001, 0.01, 0.1, 1.0]  # m² (different leaf sizes)
leaf_descriptions = ["Small leaf (1 cm²)", "Medium leaf (10 cm²)", "Large leaf (100 cm²)", "Plant canopy (1 m²)"]

println("Plant Solar Energy Conversion:")
println(repeat('-', 70))
@printf "%-20s %10s %12s %15s %15s\n" "System" "Area (cm²)" "Solar (W)" "Chemical (W)" "Daily Energy (kJ)"
println(repeat('-', 70))

for (i, area) in enumerate(leaf_areas)
    solar_power = solar_irradiance * area
    chemical_power = solar_power * photosynthesis_eff
    daily_energy = chemical_power * 8 * 3600 / 1000  # 8 hours daylight in kJ
    
    @printf "%-20s %10.1f %12.3f %15.4f %15.2f\n" leaf_descriptions[i] (area*10000) solar_power chemical_power daily_energy
end

println()
println("Photosynthesis Efficiency Analysis:")
@printf "• Average efficiency: %.1f%% (solar → chemical energy)\n" (photosynthesis_eff * 100)
println("• Theoretical maximum: ~11% (thermodynamic limit)")
println("• Loss mechanisms: reflection, non-photosynthetic absorption,")
println("  quantum yield losses, metabolic costs")
println("• Still remarkably efficient for a biological process!")
println()

# Compare to human energy needs
human_daily_energy = bmr_human * 24 * 3600 / 1000  # kJ/day
forest_area_needed = human_daily_energy / (photosynthesis_eff * solar_irradiance * 8 * 3600 / 1000)

println("Sustaining Human Life through Photosynthesis:")
@printf "• Human daily energy need: %.0f kJ/day\n" human_daily_energy
@printf "• Forest area required: %.1f m² per person\n" forest_area_needed
@printf "• That's a %.1f × %.1f meter square of forest!\n" sqrt(forest_area_needed) sqrt(forest_area_needed)
println("• Demonstrates the fundamental role of plants in ecosystems")
println()

Out[13]:

=== PHOTOSYNTHESIS: SOLAR ENERGY CONVERSION ===
Plant Solar Energy Conversion:
----------------------------------------------------------------------
System               Area (cm²)    Solar (W)    Chemical (W) Daily Energy (kJ)
----------------------------------------------------------------------
Small leaf (1 cm²)         10.0        1.000          0.0200            0.58
Medium leaf (10 cm²)      100.0       10.000          0.2000            5.76
Large leaf (100 cm²)     1000.0      100.000          2.0000           57.60
Plant canopy (1 m²)     10000.0     1000.000         20.0000          576.00

Photosynthesis Efficiency Analysis:
• Average efficiency: 2.0% (solar → chemical energy)
• Theoretical maximum: ~11% (thermodynamic limit)
• Loss mechanisms: reflection, non-photosynthetic absorption,
  quantum yield losses, metabolic costs
• Still remarkably efficient for a biological process!

Sustaining Human Life through Photosynthesis:
• Human daily energy need: 7088 kJ/day
• Forest area required: 12.3 m² per person
• That's a 3.5 × 3.5 meter square of forest!
• Demonstrates the fundamental role of plants in ecosystems

3. Modern Energy Systems and Climate Applications

Renewable Energy Thermodynamics

Solar Photovoltaic:

Theoretical maximum efficiency: ~33% (Shockley-Queisser limit)
Commercial silicon cells: ~20-22%
Multi-junction cells: >40% (concentrated sunlight)

Wind Energy:

Betz limit: Maximum 59.3% of kinetic energy can be extracted
Modern turbines: ~45-50% efficiency

Geothermal:

Efficiency limited by: Temperature difference between ground and surface
Typical efficiency: 10-15%

Energy Storage Thermodynamics

Battery Systems:

Lithium-ion: ~90-95% round-trip efficiency
Pumped hydro: ~80-85% efficiency
Compressed air: ~60-70% efficiency

Climate Thermodynamics

Earth's Energy Balance: $\text{Solar Input} = \text{Thermal Radiation Output}$

Stefan-Boltzmann for Earth: $T_{Earth} = \left(\frac{(1-\alpha)S}{4\sigma}\right)^{1/4}$

where $\alpha$ is albedo and $S$ is solar constant.

Greenhouse Effect:

Atmosphere absorbs long-wave radiation
Re-radiates partially back to surface
Effective increase in surface temperature

Carbon Cycle Thermodynamics

CO₂ dissolution in oceans:

Temperature-dependent solubility
pH changes affect carbonate equilibrium
Ocean acidification consequences

In [14]:

using Printf

println("=== RENEWABLE ENERGY THERMODYNAMICS ===")

# Solar energy analysis
function solar_cell_power(area, irradiance, efficiency)
    # Solar cell power output: P = Area × Irradiance × Efficiency
    return area * irradiance * efficiency
end

function wind_power(air_density, area, wind_speed, efficiency)
    # Wind turbine power: P = 0.5 ρ A v^3 η
    return 0.5 * air_density * area * wind_speed^3 * efficiency
end

# Solar photovoltaic analysis
println("Solar Photovoltaic Systems:")
solar_data = [
    ("Residential (5 kW)", 25, 1000, 0.20),
    ("Commercial (100 kW)", 500, 1000, 0.22),
    ("Utility scale (10 MW)", 50000, 1000, 0.24),
    ("Concentrated PV", 100, 2000, 0.40)
]

println(repeat("-", 70))
@printf "%-20s %10s %10s %10s %12s\n" "System Type" "Area (m²)" "Irrad. (W/m²)" "Efficiency" "Power (kW)"
println(repeat("-", 70))

for (system, area, irradiance, efficiency) in solar_data
    power = solar_cell_power(area, irradiance, efficiency) / 1000  # Convert to kW
    @printf "%-20s %10.0f %10.0f %10.1f%% %12.1f\n" system area irradiance (efficiency*100) power
end

println()

# Wind energy analysis
println("Wind Energy Systems:")
wind_data = [
    ("Small residential", 1.2, 5, 0.35),
    ("Medium commercial", 1.2, 8, 0.45),
    ("Large offshore", 1.2, 12, 0.50),
    ("Offshore wind farm", 1.2, 15, 0.50)
]

println(repeat("-", 80))
@printf "%-20s %12s %12s %10s %12s\n" "System Type" "Air Density" "Wind Speed" "Efficiency" "Power/m²"
println(repeat("-", 80))

for (system, air_density, wind_speed, efficiency) in wind_data
    power_per_area = wind_power(air_density, 1.0, wind_speed, efficiency)
    @printf "%-20s %12.1f %12.0f %10.1f%% %12.0f\n" system air_density wind_speed (efficiency*100) power_per_area
end

println()
println("Renewable Energy Insights:")
println("• Solar: Power scales with area and efficiency")
println("• Wind: Power scales with wind speed cubed (v^3)!")
println("• Efficiency improvements have dramatic impact")
println("• Geographic location crucial (sun/wind resources)")
println("• Storage needed for intermittent sources")
println()

# Betz limit demonstration
betz_limit = 16/27  # ≈ 0.593
println("Wind Energy Theoretical Limits:")
@printf "• Betz limit: %.1f%% (maximum extractable kinetic energy)\n" (betz_limit * 100)
println("• Modern turbines achieve ~75-85% of Betz limit")
println("• Demonstrates fundamental thermodynamic constraints")
println()

Out[14]:

=== RENEWABLE ENERGY THERMODYNAMICS ===
Solar Photovoltaic Systems:
----------------------------------------------------------------------
System Type           Area (m²) Irrad. (W/m²) Efficiency   Power (kW)
----------------------------------------------------------------------
Residential (5 kW)           25       1000       20.0%          5.0
Commercial (100 kW)         500       1000       22.0%        110.0
Utility scale (10 MW)      50000       1000       24.0%      12000.0
Concentrated PV             100       2000       40.0%         80.0

Wind Energy Systems:
--------------------------------------------------------------------------------
System Type           Air Density   Wind Speed Efficiency     Power/m²
--------------------------------------------------------------------------------
Small residential             1.2            5       35.0%           26
Medium commercial             1.2            8       45.0%          138
Large offshore                1.2           12       50.0%          518
Offshore wind farm            1.2           15       50.0%         1012

Renewable Energy Insights:
• Solar: Power scales with area and efficiency
• Wind: Power scales with wind speed cubed (v^3)!
• Efficiency improvements have dramatic impact
• Geographic location crucial (sun/wind resources)
• Storage needed for intermittent sources

Wind Energy Theoretical Limits:
• Betz limit: 59.3% (maximum extractable kinetic energy)
• Modern turbines achieve ~75-85% of Betz limit
• Demonstrates fundamental thermodynamic constraints

In [15]:

# julia
using Printf

# Stefan–Boltzmann constant (W·m⁻²·K⁻⁴)
const σ_SB = 5.670374419e-8

# Climate thermodynamics
"""Earth's equilibrium temperature: T = [(1-α)S/(4σ)]^(1/4)"""
function earth_temperature(solar_constant, albedo)
    return ((1 - albedo) * solar_constant / (4 * σ_SB))^(1/4)
end

"""Greenhouse warming effect in Kelvin"""
function greenhouse_effect(T_no_atmosphere, T_with_atmosphere)
    return T_with_atmosphere - T_no_atmosphere
end

println("=== CLIMATE THERMODYNAMICS ===")
println("Earth's Energy Balance and Greenhouse Effect")
println()

# Earth energy balance
S = 1361  # W/m² (solar constant)
albedo_earth = 0.30  # Earth's average albedo
albedo_ice = 0.90    # Ice/snow albedo
albedo_ocean = 0.06  # Ocean albedo
albedo_forest = 0.15 # Forest albedo

T_no_atmosphere = earth_temperature(S, albedo_earth)
T_current = 288.15  # K (current global average)
greenhouse_warming = greenhouse_effect(T_no_atmosphere, T_current)

println("Earth Energy Balance Analysis:")
@printf "• Solar constant: S = %.0f W/m²\n" S
@printf "• Earth's albedo: α = %.2f\n" albedo_earth
@printf "• Temperature without atmosphere: %.1f K = %.1f°C\n" T_no_atmosphere (T_no_atmosphere - 273.15)
@printf "• Current global average: %.1f K = %.1f°C\n" T_current (T_current - 273.15)
@printf "• Greenhouse effect: +%.1f K (+%.1f°C)\n" greenhouse_warming greenhouse_warming
println()

# Albedo feedback effects
println("Albedo Feedback in Climate System:")
surfaces = [
    ("Fresh snow", 0.85),
    ("Sea ice", 0.70),
    ("Desert sand", 0.35),
    ("Forest", 0.15),
    ("Grassland", 0.20),
    ("Ocean", 0.06)
]

println(repeat("-", 60))
@printf "%-15s %10s %15s %15s\n" "Surface Type" "Albedo" "T_equilibrium (K)" "T_equilibrium (°C)"
println(repeat("-", 60))

for (surface, albedo) in surfaces
    T_eq = earth_temperature(S, albedo)
    T_eq_celsius = T_eq - 273.15
    @printf "%-15s %10.2f %15.1f %15.1f\n" surface albedo T_eq T_eq_celsius
end

println()
println("Climate Feedback Insights:")
println("• Ice-albedo feedback: Melting ice → lower albedo → more warming")
println("• Forest fires increase albedo temporarily")
println("• Ocean absorption crucial for energy balance")
println("• Clouds have complex albedo and greenhouse effects")
println("• Small albedo changes cause significant temperature changes")
println()

# CO₂ and temperature relationship (simplified)
println("Greenhouse Gas Effects:")
co2_preindustrial = 280  # ppm
co2_current = 420       # ppm
co2_doubled = 560       # ppm

# Rough climate sensitivity: ΔT ≈ 3°C per CO₂ doubling
climate_sensitivity = 3.0  # K per CO₂ doubling
warming_current = climate_sensitivity * log(co2_current / co2_preindustrial) / log(2)
warming_doubled = climate_sensitivity * log(co2_doubled / co2_preindustrial) / log(2)

@printf "• Pre-industrial CO₂: %.0f ppm\n" co2_preindustrial
@printf "• Current CO₂: %.0f ppm\n" co2_current
@printf "• Current warming: +%.1f°C (from CO₂ alone)\n" warming_current
@printf "• Doubled CO₂ warming: +%.1f°C\n" warming_doubled
println("• Climate sensitivity: fundamental thermodynamic property")
println("• Logarithmic relationship: equal % increases → equal warming")
println()

Out[15]:

=== CLIMATE THERMODYNAMICS ===
Earth's Energy Balance and Greenhouse Effect

Earth Energy Balance Analysis:
• Solar constant: S = 1361 W/m²
• Earth's albedo: α = 0.30
• Temperature without atmosphere: 254.6 K = -18.6°C
• Current global average: 288.1 K = 15.0°C
• Greenhouse effect: +33.6 K (+33.6°C)

Albedo Feedback in Climate System:
------------------------------------------------------------
Surface Type        Albedo T_equilibrium (K) T_equilibrium (°C)
------------------------------------------------------------
Fresh snow            0.85           173.2           -99.9
Sea ice               0.70           206.0           -67.2
Desert sand           0.35           249.9           -23.2
Forest                0.15           267.2            -5.9
Grassland             0.20           263.2            -9.9
Ocean                 0.06           274.0             0.9

Climate Feedback Insights:
• Ice-albedo feedback: Melting ice → lower albedo → more warming
• Forest fires increase albedo temporarily
• Ocean absorption crucial for energy balance
• Clouds have complex albedo and greenhouse effects
• Small albedo changes cause significant temperature changes

Greenhouse Gas Effects:
• Pre-industrial CO₂: 280 ppm
• Current CO₂: 420 ppm
• Current warming: +1.8°C (from CO₂ alone)
• Doubled CO₂ warming: +3.0°C
• Climate sensitivity: fundamental thermodynamic property
• Logarithmic relationship: equal % increases → equal warming

4. Quantum and Nanoscale Thermodynamics

Quantum Thermodynamics

At the quantum scale, thermodynamics takes on new features:

Quantum Heat Engines:

Working medium consists of quantum states
Coherence effects can enhance efficiency
Quantum correlations play thermodynamic roles

Landauer's Principle: $E_{min} = k_B T \ln 2$

Minimum energy to erase one bit of information.

Maxwell's Demon:

Information has thermodynamic cost
Measurement and memory erasure require energy
Connects information theory to thermodynamics

Nanoscale Thermodynamics

Brownian Motion:

Thermal fluctuations dominate at small scales
Random walk behavior
Einstein-Smoluchowski relation

Molecular Motors:

Convert chemical energy to mechanical work
Operate in high-friction, low-Reynolds-number environment
Examples: ATP synthase, kinesin, myosin

Fluctuation Theorems:

Jarzynski equality
Crooks fluctuation theorem
Non-equilibrium work relations

In [16]:

using Printf

# Physical constants (SI)
const k_B = 1.380649e-23       # Boltzmann constant (J/K)
const h   = 6.62607015e-34     # Planck constant (J·s)

println("=== QUANTUM AND NANOSCALE THERMODYNAMICS ===")

# Landauer's principle
function landauer_energy(T)
    # minimum energy to erase one bit
    return k_B * T * log(2)
end

# Thermal de Broglie wavelength: λ = h/√(2π m k_B T)
function thermal_de_broglie(T, mass)
    return h / sqrt(2π * mass * k_B * T)
end

# Einstein relation: D = k_B T / (6π η r) for spherical particle
function diffusion_coefficient(T, radius, viscosity)
    return k_B * T / (6π * viscosity * radius)
end

println("=== INFORMATION THERMODYNAMICS ===")
println("Landauer's Principle: Energy Cost of Information Processing")
println()

temperatures = [4, 77, 300, 373]  # K (liquid helium, liquid N₂, room temp, boiling water)
temp_names = ["Liquid He", "Liquid N₂", "Room temp", "Boiling H₂O"]

println("Minimum Energy to Erase One Bit (Landauer Limit):")
println(repeat("-", 60))
@printf "%-15s %8s %15s %15s\n" "Temperature" "T (K)" "Energy (J)" "Energy (eV)"
println(repeat("-", 60))

for (i, T) in enumerate(temperatures)
    E_landauer = landauer_energy(T)
    E_eV = E_landauer / 1.602176634e-19  # Convert to eV
    @printf "%-15s %8.0f %15.2e %15.2e\n" temp_names[i] T E_landauer E_eV
end

println()
println("Information Processing Insights:")
println("• Colder computers are more energy-efficient")
println("• Quantum computers cooled to millikelvin temperatures")
println("• Modern processors dissipate ~10⁶ times Landauer limit")
println("• Room for dramatic improvement in reversible computing")
println()

# Quantum vs classical regimes
println("=== QUANTUM VS CLASSICAL REGIMES ===")
println("Thermal de Broglie Wavelength")
println()

particles = [
    ("Electron", 9.109e-31),
    ("Proton", 1.673e-27),
    ("H₂ molecule", 3.3e-27),
    ("Virus (100 nm)", 1e-15),
    ("Dust (1 μm)", 1e-12)
]

T_room = 300  # K

println("Thermal de Broglie wavelengths at T = $T_room K:")
println(repeat("-", 70))
@printf "%-15s %12s %15s %15s\n" "Particle" "Mass (kg)" "λ_thermal (m)" "Quantum?"
println(repeat("-", 70))

for (particle, mass) in particles
    λ_th = thermal_de_broglie(T_room, mass)
    quantum_regime = λ_th > 1e-10 ? "Yes" : "No"  # Rough criterion
    @printf "%-15s %12.2e %15.2e %15s\n" particle mass λ_th quantum_regime
end

println()
println("Quantum Regime Criteria:")
println("• Thermal wavelength comparable to system size")
println("• Light particles more likely to show quantum effects")
println("• Lower temperatures enhance quantum behavior")
println("• Macroscopic objects are classical")
println()

# Nanoscale Brownian motion
println("=== NANOSCALE BROWNIAN MOTION ===")
η_water = 1e-3  # Pa⋅s (water viscosity)
T_body = 310    # K (body temperature)

nano_objects = [
    ("Small molecule", 1e-9),
    ("Protein", 5e-9),
    ("Virus", 50e-9),
    ("Bacterium", 1e-6),
    ("Cell", 10e-6)
]

println("Diffusion in Water at Body Temperature:")
println(repeat("-", 70))
@printf "%-15s %12s %15s %15s\n" "Object" "Radius (m)" "D (m²/s)" "Distance/s (μm)"
println(repeat("-", 70))

for (object, radius) in nano_objects
    D = diffusion_coefficient(T_body, radius, η_water)
    distance_per_sec = sqrt(2 * D * 1) * 1e6  # RMS distance in 1 second, in μm
    @printf "%-15s %12.2e %15.2e %15.1f\n" object radius D distance_per_sec
end

println()
println("Nanoscale Transport Insights:")
println("• Smaller objects diffuse faster")
println("• Thermal motion dominates at nanoscale")
println("• Molecular motors work against Brownian motion")
println("• Diffusion-limited reaction rates")
println("• No Reynolds number effects (Stokes flow)")
println()

Out[16]:

=== QUANTUM AND NANOSCALE THERMODYNAMICS ===
=== INFORMATION THERMODYNAMICS ===
Landauer's Principle: Energy Cost of Information Processing

Minimum Energy to Erase One Bit (Landauer Limit):
------------------------------------------------------------
Temperature        T (K)      Energy (J)     Energy (eV)
------------------------------------------------------------
Liquid He              4        3.83e-23        2.39e-04
Liquid N₂             77        7.37e-22        4.60e-03
Room temp            300        2.87e-21        1.79e-02
Boiling H₂O          373        3.57e-21        2.23e-02

Information Processing Insights:
• Colder computers are more energy-efficient
• Quantum computers cooled to millikelvin temperatures
• Modern processors dissipate ~10⁶ times Landauer limit
• Room for dramatic improvement in reversible computing

=== QUANTUM VS CLASSICAL REGIMES ===
Thermal de Broglie Wavelength

Thermal de Broglie wavelengths at T = 300 K:
----------------------------------------------------------------------
Particle           Mass (kg)   λ_thermal (m)        Quantum?
----------------------------------------------------------------------
Electron            9.11e-31        4.30e-09             Yes
Proton              1.67e-27        1.00e-10             Yes
H₂ molecule         3.30e-27        7.15e-11              No
Virus (100 nm)      1.00e-15        1.30e-16              No
Dust (1 μm)         1.00e-12        4.11e-18              No

Quantum Regime Criteria:
• Thermal wavelength comparable to system size
• Light particles more likely to show quantum effects
• Lower temperatures enhance quantum behavior
• Macroscopic objects are classical

=== NANOSCALE BROWNIAN MOTION ===
Diffusion in Water at Body Temperature:
----------------------------------------------------------------------
Object            Radius (m)        D (m²/s) Distance/s (μm)
----------------------------------------------------------------------
Small molecule      1.00e-09        2.27e-10            21.3
Protein             5.00e-09        4.54e-11             9.5
Virus               5.00e-08        4.54e-12             3.0
Bacterium           1.00e-06        2.27e-13             0.7
Cell                1.00e-05        2.27e-14             0.2

Nanoscale Transport Insights:
• Smaller objects diffuse faster
• Thermal motion dominates at nanoscale
• Molecular motors work against Brownian motion
• Diffusion-limited reaction rates
• No Reynolds number effects (Stokes flow)

5. CoCalc for Modern Thermodynamics Research

Interdisciplinary Collaboration

Modern thermodynamics requires collaboration across multiple fields:

Astrophysics Teams:

Theoretical physicists modeling stellar evolution
Observational astronomers analyzing stellar spectra
Computational experts running stellar structure simulations
Real-time collaboration on complex calculations and data analysis

Biophysics Research:

Molecular biologists understanding protein folding
Biochemists measuring metabolic energetics
Physicists developing theoretical models
Live sharing of experimental data and theoretical predictions

Climate Science:

Atmospheric physicists modeling energy balance
Oceanographers studying thermal transport
Climate modelers integrating complex systems
Policy researchers analyzing mitigation strategies

Advanced Computational Tools

Julia for Scientific Computing:

High-performance numerical calculations
Excellent package ecosystem for physics and biology
Seamless integration with visualization libraries
Collaborative development of research codes

Real-Time Mathematical Typesetting:

LaTeX equations: $S = k_B \ln \Omega$ , $L = 4\pi R^2 \sigma T_{eff}^4$
Professional-quality scientific documents
Immediate feedback on mathematical derivations
Version control for tracking theoretical developments

Best Practices for Modern Research

Scientific Rigor:

Document all assumptions and approximations clearly
Validate models against experimental data
Perform uncertainty analysis on key results
Cross-check calculations using multiple approaches

Reproducible Science:

Version control for all code and data
Clear documentation of computational methods
Open sharing of analysis workflows
Collaborative peer review processes

Impact and Communication:

Connect fundamental physics to real-world applications
Visualize complex thermodynamic relationships
Engage with broader scientific community
Translate research for public understanding

Modern thermodynamics continues to expand our understanding of the universe, from the quantum realm to cosmic scales. CoCalc provides the collaborative tools needed to tackle these complex, interdisciplinary challenges and make breakthrough discoveries at the frontiers of science.