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Today in History – 24 May

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today in history 24 may

today in history 24 may

1543

On May 24, 1543, Polish astronomer Nicolaus Copernicus died in what is now Frombork, Poland. The father of modern astronomy, he was the first modern European scientist to propose that Earth and other planets revolve around the sun.

1557

Sikandar Sur was compelled to surrender Mankot in Punjab.

1775

On this day in 1775, John Hancock was elected president of the Second Continental Congress. John Hancock was best known for his large signature on the Declaration of Independence, which he jested the British could read without spectacles. He was serving as president of Congress upon the declaration’s adoption on July 4, 1776, and, as such, was the first member of the Congress to sign the historic document.

1813

Krishna Mohan Bandopadhyay, leader, essay writer and journalist in Bengal, was born.

1875

Sir Sayed Ahmed Khan established Mohmmad Anglo Oriental School which was renamed in 1920 as ‘Aligarh Muslim University’.

1883

After 14 years and 27 deaths while being constructed, the Brooklyn Bridge over the East River was opened, connecting the great cities of New York and Brooklyn for the first time in history. Thousands of residents of Brooklyn and Manhattan Island turned out to witness the dedication ceremony, which was presided over by President Chester A. Arthur and New York Governor Grover Cleveland. Designed by the late John A. Roebling, the Brooklyn Bridge was the largest suspension bridge ever built to that date.

1935

The Cincinnati Reds beat the Philadelphia Phillies 2-1 on this night in 1935 in Major League Baseball’s first-ever night game, played courtesy of recently installed lights at Crosley Fiel in Cincinnati.

1940

Today is the birthday of poet Joseph Brodsky, born this day in St. Petersburg, Russia. His poetry treats such universal topics as life, death, and the meaning of existence. Brodsky’s early poetry won critical acclaim, but the Soviet government considered him a loafer and sentenced him to five years of hard labor for “social parasitism.”

1956

Gautam Buddha‘s 2500th birth anniversary was celebrated.

1960

Dr. Ida Sophie Schruder, founder of famous ‘Vellore Hospital’, passed away in Kodaikanal.

1964

A referee’s call in a soccer match between Peru and Argentina sparks a riot on this day in 1964. More than 300 fans were killed and another 500 people were injured in the violent melee that followed at National Stadium in Lima, Peru.

1990

K. S. Hegde, former President of Parliament, passed away.

1991

The body of Rajiv Gandhi, India’s assassinated former premier and son of the late Indira Gandhi, was cremated in New Delhi today. He was killed by a suicide bomber in the southern state of Tamil Nadu three days ago. Police suspect that Tamil rebels, fighting for independence in Sri Lanka, carried out the murder. Gandhi’s death signaled the end of the Nehru dynasty’s rule over India. His two children were too young to assume the leadership of this turbulent nation. There was pressure for Sonia Gandhi to succeed her husband, but she was Italian-born and refused.

1993

India and Uzbekistan signed five pacts.

2000

The Supreme Court rejects the proposal for setting up benches outside Delhi.

Related Articles:

Thermodynamics and Entropy: Foundations of Irreversibility and Energy Flow

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thermodynamics and entropy

Table of Contents

  1. Introduction
  2. What Is Thermodynamics?
  3. The Four Laws of Thermodynamics
  4. Internal Energy and the First Law
  5. Work, Heat, and State Functions
  6. The Concept of Entropy
  7. Entropy and Reversible vs Irreversible Processes
  8. The Second Law and the Arrow of Time
  9. Entropy in Statistical Mechanics
  10. Thermodynamic Potentials
  11. Maxwell’s Relations
  12. Real-World Applications
  13. Conclusion

1. Introduction

Thermodynamics is the science of energy transformations and the relationships between heat, work, and internal energy. It is both a phenomenological and fundamental science, underpinning engines, phase changes, chemical reactions, and even black holes.

At the heart of thermodynamics lies entropy — a concept that bridges physics, probability, and the arrow of time.

2. What Is Thermodynamics?

Thermodynamics studies the macroscopic properties of matter — temperature, pressure, volume, and energy — without needing microscopic detail.

Key questions include:

  • How does energy flow?
  • What processes are possible or forbidden?
  • Why do systems evolve toward equilibrium?

3. The Four Laws of Thermodynamics

Zeroth Law (Thermal Equilibrium):

If \( A = B \) and \( B = C \) are in thermal equilibrium, then \( A = C \).

This allows the definition of temperature as a measurable, transitive quantity.

First Law (Energy Conservation):

\[
\Delta U = Q – W
\]

Change in internal energy equals heat added minus work done by the system.

Second Law (Entropy Increase):

In any spontaneous process:

\[
\Delta S_{\text{universe}} \geq 0
\]

Entropy tends to increase, leading to irreversibility.

Third Law (Zero Entropy at Absolute Zero):

As \( T \rightarrow 0 \), \( S \rightarrow 0 \) for a perfect crystal.


4. Internal Energy and the First Law

Internal energy \( U \) is the total microscopic energy of a system (kinetic + potential of atoms).

The first law states:

\[
dU = \delta Q – \delta W
\]

Where:

  • \(delta Q\): infinitesimal heat input
  • \(delta W\): infinitesimal work output

5. Work, Heat, and State Functions

  • Work and heat are path-dependent (not state functions)
  • Internal energy, entropy, and volume are state functions

For example, in a quasistatic expansion:

\[
\delta W = P dV
\]

The total work done depends on the path taken in P-VP\text{-}VP-V space.


6. The Concept of Entropy

Entropy SSS quantifies disorder, multiplicity of microstates, or unavailable energy.

Clausius defined:

\[
dS = \frac{\delta Q_{\text{rev}}}{T}
\]

For a reversible process.

Entropy is a state function — its change depends only on initial and final states, not the path.


7. Entropy and Reversible vs Irreversible Processes

For Reversible processes :

\[
\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}
\]

For Irreversible (real) processes:

\[
\Delta S > \int \frac{\delta Q}{T}
\]

Entropy increases in spontaneous processes — never decreases for isolated systems.


8. The Second Law and the Arrow of Time

The second law introduces time asymmetry — systems evolve from less probable to more probable states.

Entropy gives a direction to time. Systems evolve toward higher entropy states:

  • Ice melts
  • Gases expand
  • Energy dissipates

These processes are irreversible, even though microscopic laws are time-symmetric.


9. Entropy in Statistical Mechanics

Boltzmann connected entropy to microstates:

\[
S = k_B \ln \Omega
\]

Where:

  • \(Omega\): number of microstates corresponding to a macrostate
  • \(k_B\)​: Boltzmann constant

This formulation bridges classical thermodynamics with statistical physics.


10. Thermodynamic Potentials

Useful for systems with constraints (e.g., constant pressure, temperature).

  • Internal energy: \( U = U(S, V) \)
  • Enthalpy: \( H = U + PV \) (constant pressure processes)
  • Helmholtz Free Energy: \( F = U – TS \) (constant temperature and volume)
  • Gibbs Free Energy: \( G = H – TS \) (constant pressure and temperature)

Each potential is minimized under different constraints, in other words systems minimize appropriate potentials in equilibrium.


11. Maxwell’s Relations

Derived from thermodynamic potentials via Legendre transformations:

Example:

\[
\left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V
\]

These provide useful identities between thermodynamic variables.

There are four such relations. They relate otherwise hard-to-measure quantities using measurable ones.


12. Real-World Applications

Thermodynamics and entropy are central in:

  • Heat engines and refrigerators (Carnot cycle)
  • Chemical reactions (Gibbs free energy and spontaneity)
  • Black hole entropy and thermodynamics
  • Biological processes (entropy production, ATP hydrolysis)
  • Information theory (Shannon entropy is thermodynamic entropy analog)

13. Conclusion

Thermodynamics provides deep insights into the behavior of matter and energy. Entropy, once viewed as abstract, is now seen as the fundamental driver of irreversible processes and the bridge between order and chaos.

Understanding entropy is not only key to mastering thermodynamics, but also to appreciating the deep structure of the universe — from steam engines to black holes to information theory.


.

Today in History – 23 May

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today in history 23 may

today in history 23 may

1701

At London’s Execution Dock, British privateer William Kidd, popularly known as Captain Kidd, was hanged for piracy and murder.

1805

Lord Wellesley, Governer General of India, issued a permanent provision for the Delhi Mughal Emperor.

1810

Writer and editor Margaret Fuller, who inspired other Americans to devote themselves to learning, was born on this day.

1895

Shivprasad Sitar-E-Hind passed away.

1900

Sergeant William Harvey Carney was awarded the Congressional Medal of Honor for his bravery on July 18, 1863, while fighting for the Union’s cause as a member of the 54th Massachusetts Colored Infantry. He was the first African American to receive the Medal of Honor, which was the nation’s highest military honor.

1903

Gemini Roy, famous Painter, joined Government School of Arts and Crafts at Calcutta.

1911

In a ceremony presided over by President William Howard Taft, the New York Public Library, the largest marble structure ever constructed in the United States, was dedicated in New York City. Occupying a two-block section of Fifth Avenue between 40th and 42nd Streets, the monumental beaux-arts structure took 14 years to complete at a cost of $9 million. The day after its dedication, the library opened its doors to the public, and some 40,000 citizens passed through to make use of a collection that already consisted of more than a million books.

1914

The British India Steamship Co. and the Peninsular and Oriental Steamship Co. announced their amalgamation at London.

1915

On this day in 1915, Italy declared war on Austria-Hungary, entering World War I on the side of the Allies—Britain, France and Russia. When World War I broke out in the summer of 1914, Italy declared itself neutral in the conflict, despite its membership in the so-called Triple Alliance alongside Germany and Austria-Hungary since 1882.

1931

Sharad Joshi, famous Hindi poet, playwright and journalist, was born.

1947

The U.K. Cabinet today took the historic step of agreeing to Lord Mountbatten’s proposal for the partition of India into two states, one Muslim and the other Hindu. The Viceroy was to have a series of talks with Lord Listowel, the Secretary for India.

1967

South Gujrat University was established.

1975

P. S. Bapat, victoria cross winner Lieutenant General, passed away.

1984

Bachendri Pal, an employee of Tata Iron and Steel Co. Limited at Jamshedpur, conquered the summit of Mount Everest to become the first Indian woman and the fifth women in the world to achieve this feat.

1995

TADA expires and new Anti-Terrorists Activities Law proposed. Criminal Law Amendment Bill, 1995, voting put off; Lok Sabha passes Finance Bill 1995.

1996

1995-96 deficit touched a new high at Rs.19,855 crore.

1949

The Federal Republic of Germany (popularly known as West Germany) was formally established as a separate and independent nation. This action marked the effective end to any discussion of reuniting East and West Germany.

1960

A tsunami caused by an earthquake off the coast of Chile traveled across the Pacific Ocean and killed 61 people in Hilo, Hawaii, on this day in 1960. The massive 8.5-magnitude quake had killed thousands in Chile the previous day.

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Classical Statistical Mechanics: Bridging Microscopic Dynamics and Macroscopic Laws

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classical statistical mechanics

Table of Contents

  1. Introduction
  2. What Is Classical Statistical Mechanics?
  3. The Need for a Statistical Approach
  4. Microstates, Macrostates, and Phase Space
  5. Liouville’s Theorem and Probability Conservation
  6. Ensembles: Microcanonical, Canonical, and Grand Canonical
  7. The Boltzmann Distribution
  8. Partition Function and Thermodynamics
  9. Equipartition Theorem
  10. Entropy and the Boltzmann H-Theorem
  11. Classical vs Quantum Statistical Mechanics
  12. Applications of Classical Statistical Mechanics
  13. Conclusion

1. Introduction

Classical statistical mechanics provides the crucial link between microscopic laws of motion and macroscopic thermodynamic behavior. It does so by applying probability theory to systems of many particles, treating the precise microstate as unknown or irrelevant, and instead focusing on averages and distributions.

This framework explains how deterministic Newtonian motion gives rise to temperature, pressure, entropy, and other thermodynamic quantities.


2. What Is Classical Statistical Mechanics?

Classical statistical mechanics describes large systems of particles using probability distributions in phase space, rather than tracking each particle’s exact trajectory.

While Newton’s laws can describe individual particles, they are impractical for \( N \sim 10^{23} \) particles in a gas. Instead, we study how macroscopic observables emerge from an ensemble of microstates.


3. The Need for a Statistical Approach

Even though the laws of classical mechanics are deterministic, our ignorance of initial conditions and the complexity of many-body systems necessitate a statistical description.

Goals:

  • Predict average properties like pressure, energy, entropy
  • Derive thermodynamic laws from Newtonian mechanics
  • Understand irreversibility and time’s arrow from reversible laws

4. Microstates, Macrostates, and Phase Space

  • Microstate: A specific configuration of positions and momenta for all particles.
  • Macrostate: A set of microstates consistent with observed macroscopic quantities (e.g., total energy, volume, number of particles).

For a system of \( N \) particles:

\[
\text{Microstate} = \{q_1, …, q_N, p_1, …, p_N\}
\]

Phase space has \( 6N \) dimensions: each microstate is a point in this space.


5. Liouville’s Theorem and Probability Conservation

Liouville’s theorem ensures that probability is conserved in phase space evolution:

\[
\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho, H\} = 0
\]

Where \( \rho(q, p, t) \) is the distribution function.

This means the density of representative points in phase space remains constant along the trajectory.


6. Ensembles: Microcanonical, Canonical, and Grand Canonical

An ensemble is a conceptual collection of many copies of the system, each in a different microstate but consistent with the same macroscopic constraints.

a. Microcanonical Ensemble

  • Isolated system (fixed \( N, V, E \))
  • All accessible microstates are equally probable

\[
\rho = \frac{1}{\Omega(E)} \quad \text{if } H = E
\]

b. Canonical Ensemble

  • System in thermal contact with a reservoir at temperature \( T \)
  • Fixed \( N, V, T \)

\[
P(E) = \frac{e^{-\beta E}}{Z}, \quad \beta = \frac{1}{k_B T}
\]

c. Grand Canonical Ensemble

  • Variable particle number
  • Used for systems exchanging particles and energy
  • Useful in quantum gases and open systems

7. The Boltzmann Distribution

In the canonical ensemble, the probability of a system being in a microstate with energy \(E_i\)​ is:

\[
P_i = \frac{e^{-\beta E_i}}{Z}
\]

This distribution reflects that higher energy states are exponentially less probable at fixed temperature.


8. Partition Function and Thermodynamics

The partition function encodes thermodynamic quantities:

\[
Z = \sum_i e^{-\beta E_i}
\]

  • Free energy: \( F = -k_B T \ln Z \)
  • Internal energy: \( U = -\frac{\partial \ln Z}{\partial \beta} \)
  • Entropy: \( S = -\frac{\partial F}{\partial T} \)
  • Pressure: \( P = -\left( \frac{\partial F}{\partial V} \right)_T \)

9. Equipartition Theorem

Each quadratic degree of freedom contributes:

\[
\langle E \rangle = \frac{1}{2} k_B T
\]

E.g., an ideal gas with \( 3N \) degrees of freedom:

\[
U = \frac{3}{2} N k_B T
\]


10. Entropy and the Boltzmann H-Theorem

Boltzmann’s entropy formula:

\[
S = k_B \ln \Omega
\]

Where \(OmegaΩ\) is the number of accessible microstates.

The H-theorem shows:

\[
\frac{dH}{dt} \leq 0
\]

showing that systems evolve towards equilibrium, justifying the second law of thermodynamics statistically.


11. Classical vs Quantum Statistical Mechanics

  • Classical: Continuous phase space, distinguishable particles
  • Quantum: Discrete states, indistinguishable particles (Bose-Einstein, Fermi-Dirac)

Classical statistical mechanics is valid when quantum effects are negligible (e.g., high temperature, low density).


12. Applications of Classical Statistical Mechanics

  • Ideal and real gases
  • Thermodynamic identities
  • Chemical reactions and equilibria
  • Thermal conductivity and diffusion
  • Statistical derivation of pressure and temperature

It provides a microscopic foundation for the laws of thermodynamics.

13. Conclusion

Classical statistical mechanics elegantly unifies deterministic laws and probabilistic reasoning. It shows how the unpredictable behavior of individual particles leads to highly predictable macroscopic behavior — all via the structure of phase space and the power of statistical ensembles.

Understanding this subject is essential for anyone working in thermodynamics, kinetic theory, or the transition to quantum and statistical field theories.

.

Today in History – 22 May

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today in history 22 may

today in history 22 may

1545

Sher Shah Sur died. He was fatally wounded in an explosion on Kalanjara fort while preparing for an attack. He introduced the new silver rupee-coin “Rupiya” based on ratio of 40 copper coin pieces (paisa) per rupee and built several roads including the longest road of India known as Grand Trunk Road (now Natioinal Highway).

1772

Raja Ram Mohan Roy, great social reformer, lawyer and politician and founder of Brahmo Samaj, was born at Radhanagar in Hooghly district, Bengal. (1771 or 1772)

1843

A massive first major wagon train, made up of 1,000 settlers and 1,000 head of cattle, sets off down the Oregon Trail from Independence, Missouri. Known as the “Great Emigration,” the expedition came two years after the first modest party of settlers made the long, overland journey to Oregon.

1859

It’s the birthday of Sir Arthur Conan Doyle, the creator of master sleuth Sherlock Holmes.

1917

Suniti Chaudhary, revolutionary freedom fighter, was born at Kumilla. She was one of the accused in the famous Chargaon ammunition plot and assassination of General Lemen, Inspector General of Dhaka.

1936

Lord Brabourne laid the foundation stone of Brabourne Stadium in Bombay.

1939

On this day in 1939, Italy and Germany agreed to a military and political alliance, giving birth formally to the Axis powers, which will ultimately included Japan. Mussolini coined the nickname “Pact of Steel” (he had also come up with the metaphor of an “axis” binding Rome and Berlin) after reconsidering his first choice, “Pact of Blood,” to describe this historic agreement with Germany.

1940

Erapalli Anantrao Srinivasarao Prasanna, cricketer (one of India’s big four spinners, Right-arm Off-break Bowler), was born at Bangalore. He was also the receipent of Arjun Award (1968) and Padmashree (1970).

1953

Sherpa Tenzing Norgay of Darjeeling and Edmund Hillary of New Zealand were the first to conquer the Mount Everest (World).

1961

Jnanpeeth Award was instituted and the first award was given in 1965. This award is given for the best creative literary writing by any Indian citizen in any of the languages included in the VIII Schedule of the Indian Constitution. The award carries a cash price of Rs 2.5 lakh, a citation and a bronze momento.

1963

Rohini, Glider, became the first to successfully fly at Kanpur.

1970

Vijayanand Patnaik established political party `Utkar Congress’.

1989

IRBM ‘Agni’ Missile launched successfully from Chandipur, Orissa.

1990

After 150 years apart, Marxist South Yemen and conservative North Yemen were unified as the Republic of Yemen. Ali Abdullah, president of North Yemen, became the new country’s president, and Ali Salem Al-Baidh, leader of the South Yemeni Socialist Party, vice president. The first free elections were held in 1993.

1992

India launched its ‘Agni’ rocket.

1996

United Front prime ministerial candidate H.D. Deve Gowda unanimously was elected leader of the Front’s parliamentary party.

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Today in History – 18 May